{"id":2833,"date":"2023-10-03T13:56:54","date_gmt":"2023-10-03T13:56:54","guid":{"rendered":"http:\/\/localhost\/ecole9ja\/?p=2833"},"modified":"2023-10-03T14:03:22","modified_gmt":"2023-10-03T14:03:22","slug":"week-1-ss2-first-term-further-mathematics-notes","status":"publish","type":"post","link":"https:\/\/ecolebooks.com\/nigeria\/posts\/week-1-ss2-first-term-further-mathematics-notes\/","title":{"rendered":"Week 1 – SS2 First Term Further Mathematics Notes"},"content":{"rendered":"

SUBJECT: FURTHER MATHEMATICS\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0CLASS: SS2
\n<\/strong>
\n\u00a0FIRST TERM SCHEME OF WORK
\n<\/strong>
\n\u00a0<\/p>\n

\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n
WEEK<\/strong><\/td>\nTOPIC<\/strong><\/td>\n<\/tr>\n
1<\/td>\nFinding quadratic equation with given sum and product of roots, conditions for equal roots, real roots and no root<\/td>\n<\/tr>\n
2<\/td>\nTangents and Normals to Curves<\/td>\n<\/tr>\n
3<\/td>\nPolynomials ;definition, basic operations + , x , – , ;– <\/td>\n<\/tr>\n
4<\/td>\nPolynomials ( Continued) factorization<\/td>\n<\/tr>\n
5<\/td>\nCubic Equation , roots of cubic equations<\/td>\n<\/tr>\n
6<\/td>\nReview and Test<\/td>\n<\/tr>\n
7<\/td>\nLogical Reasoning ; fundamental issues and definitions and theorem proving<\/td>\n<\/tr>\n
8<\/td>\nTrigonometric Function , six trig functions of angles of any magnitude ( sine, cosine,tangent,secant, cosecant, cotangent)<\/td>\n<\/tr>\n
9<\/td>\nRelationship between graph of trigonometric ratios such as sin x and sin 2x, graphs of y= a sin (bx) + c , y = a cos (bx) + c , y = a tan (bx) + c<\/td>\n<\/tr>\n
10<\/td>\nGraphs of inverse by ratio and equation of simpletrgonometric identities<\/td>\n<\/tr>\n
11<\/td>\nRevision<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n

\u00a0REFERENCES
\n<\/strong><\/p>\n

    \n
  • Further Mathematics Project 1 by TuttuhAdegun\n<\/li>\n
  • Further Mathematics Project 2 by TuttuhAdegun\n<\/li>\n
  • Additional Mathematics by Godman\n<\/li>\n<\/ul>\n

    \u00a0WEEK 1
    \n<\/strong>TOPIC: SOLUTION TO QUADRATIC EQUATION
    \n<\/strong>FINDING QUADRATIC EQUATION GIVEN SUM AND PRODUCT OF ROOTS CONDITION FOR EQUAL ROOTS, REAL ROOTS AND NO ROOT
    \n<\/strong>
    \n\u00a0We recall that if ax2<\/sup> + bx + c = 0, where a, a and c are constants such that a \u2260 0, then,
    \n<\/strong>x = or x = <\/p>\n

    \u00a0Suppose we represent these distinct roots by \u03b1 and \u03b2; thus:
    \n<\/strong>\u03b1 =
    \nand
    \n\u03b2<\/p>\n

    \u00a0We may also put D = b2<\/sup> \u2013 4ac, so that
    \n<\/strong>\u03b1=
    \n\u03b2 = <\/p>\n

    \u00a0Sum of roots
    \n<\/strong>\u03b1 + \u03b2 = +
    \n= <\/p>\n

    \u00a0=
    \nProducts of roots
    \n<\/strong>\u03b1\u03b2 =
    \n\u02f8\u03b1\u03b2 = b2<\/sup> \u2013 D
    \n\"\"\/\u00a0\u00a0\u00a0\u00a04a2<\/sup>
    \n\t\t= b2<\/sup> \u2013 (b2<\/sup> \u2013 4ac)
    \n\"\"\/\u00a0\u00a0\u00a0\u00a0 4a2<\/sup>
    \n\t\t= 4ac
    \n\"\"\/ 4a2<\/sup>
    \n\t\t=
    \nHence, if ax2<\/sup> + bx + c = 0, where a, b and c are constants and\u03b1\u2260 0 then \u03b1 + \u03b2= ,
    \n\u03b1\u03b2 =
    \nx2<\/sup> + x\u2013 42 = 0
    \nthen (x \u2013 6) (x \u2013 7) = 0<\/p>\n

    \u00a0Hence the roots of the equation are 6 and -7. In general, if a quadratic equation factorizes into
    \n<\/strong>(x \u2013 \u03b1) (x – \u03b2) = 0
    \nthen \u03b1 and \u03b2 must be the roots of that equation. <\/p>\n

    \u00a0The general quadratic equation ax2<\/sup> + bx + c = 0 can also be written as:
    \nx2<\/sup> + \u2026(1)<\/p>\n

    \u00a0If the roots of the equation are \u03b1 and \u03b2 then the above equation can be written as:
    \n<\/strong>(x \u2013\u03b1) (x \u2013 \u03b2) = 0
    \nx2<\/sup> \u2013 (\u03b1 \u2013 \u03b2) x + \u03b1\u03b2 = 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0—(2
    \nBy comparing coefficients in equations (1) and (2)
    \n-(\u03b1 + \u03b2) =
    \n: \u03b1 + \u03b2 =
    \nand\u03b1\u03b2 =
    \nThe above consideration gives rise to two problems:
    \n(a) Given a quadratic equation, we can find the sum and product of the roots.
    \n(b) Given the roots, we can formulate the corresponding quadratic equation.
    \nThe quadratic equation whose roots are \u03b1 and \u03b2 is
    \nx2<\/sup> \u2013 (\u03b1 + \u03b2) x + \u03b1 \u03b2 = 0<\/p>\n

    \u00a0Find the sum and product of the roots of each of the following quadratic equations:
    \n<\/strong>(a) 2x2<\/sup> + 3x \u2013 1 = 0
    \n(b) 3x2<\/sup> \u2013 5x \u2013 2 = 0
    \n(c) x2<\/sup> \u2013 4x \u2013 3 = 0
    \n(d) \u00bd x2<\/sup> \u2013 3x \u2013 1 = 0<\/p>\n

    \u00a0Solution
    \n<\/strong>(a) 2x2<\/sup> + 3x \u2013 1 = 0
    \na = 2; b = 3; c = -1
    \nLet \u03b1 and \u03b2 be the roots of the equation, then
    \n\u03b1 + \u03b2=
    \n\u03b1 \u03b2 =
    \n(b) 3x2<\/sup> \u2013 5x \u2013 2 = 0
    \na = 3; b = -5; c = -2
    \nLet \u03b1 and \u03b2 be the root of the equation, then
    \n\u03b1 + \u03b2 =
    \n\u03b1 \u03b2 =
    \n(c) x2<\/sup> \u2013 4x \u2013 3 = 0
    \na = 1; b = 4; c = -3
    \nLet \u03b1 and \u03b2 be the root of the equation, then
    \n\u03b1 + \u03b2 =
    \n\u03b1 \u03b2 =
    \n(d) \u00bd x2<\/sup> \u2013 3x \u2013 1 = 0
    \na = \u00bd, b = -3, c = -1
    \nLet \u03b1 and \u03b2 be the root of the equation, then
    \n\u03b1 + \u03b2 =
    \n\u03b1 \u03b2 = = -2<\/p>\n

    \u00a0Find the quadratic equation whose roots are:
    \n<\/strong>(a) 3 and -2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) \u00bd and 5
    \n(c) -1 and 8\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(d)\u00be and \u00bd <\/p>\n

    \u00a0Solution
    \n<\/strong>The quadratic equation whose roots are \u03b1 and \u03b2 is x2<\/sup> \u2013 (\u03b1 + \u03b2) x +\u03b1 \u03b2 = 0.
    \n(a) \u03b1 + \u03b2 = 3 \u2013 2 = 1, \u03b1 \u03b2 = 3 (-2) = -6
    \n: The quadratic equation whose roots are 3 and -2 is x2<\/sup> \u2013 x \u2013 6 = 0.<\/p>\n

    \u00a0(b) \u03b1 \u03b2 = \u03b1 \u03b2 =
    \n:The quadratic equation whose roots are
    \nx2<\/sup>\u2013
    \nor 2x2<\/sup> \u2013 11x + 5 = 0<\/p>\n

    \u00a0(c) \u03b1+ \u03b2 = 7, \u03b1 \u03b2 = -8
    \n:\u03b1 \u03b2 = 7,\u03b1 \u03b2 = -8
    \n:The quadratic equation whose roots are -1 and 8 is x2<\/sup> \u2013 7x \u2013 8 = 0.<\/p>\n

    \u00a0(b) \u03b1+ \u03b2 = \u03b1 \u03b2 =
    \n:The quadratic equation whose roots are \u00be and \u00bd is
    \nx2<\/sup>\u2013
    \nor 8x2<\/sup> \u2013 10x + 3 = 0<\/p>\n

    \u00a0Symmetric Properties of Roots
    \n<\/strong>of ax2<\/sup> + bx + c = 0, then
    \n\u03b1 + \u03b2 = \u03b1 \u03b2 =
    \nCertain relations involving \u03b1 and \u03b2 can also be determined from \u03b1 + \u03b2 and \u03b1 \u03b2 even when we do not know\u03b1 and \u03b2 distinctively. Such relations are usually said to be symmetric.
    \nThey are symmetric in the sense that if \u03b1 and \u03b2 are interchanged, either the relation remains the same or is multiplied by -1.
    \nIf \u03b1\u2260 \u03b2, determine whether or not each of the following is symmetric:
    \n(a) \u03b1 + \u03b2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) \u03b1\u03b2
    \n(c) \u03b12<\/sup> \u03b22<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(d) \u03b12<\/sup>\u2013 \u03b22<\/sup>
    \n\t\t(e) 3\u03b1 +2\u03b2 \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(f) \u03b12<\/sup> \u03b22<\/sup><\/p>\n

    \u00a0Solution
    \n<\/strong>(a) \u03b1+ \u03b2 =\u03b2 + \u03b1
    \n: \u03b1 + \u03b2 is symmetric<\/p>\n

    \u00a0(b)\u03b1\u03b2 = \u03b2\u03b1
    \n: \u03b1\u03b2 is symmetric<\/p>\n

    \u00a0(c) \u03b12<\/sup> \u03b22<\/sup>= \u03b12<\/sup> \u03b22<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n: \u03b12<\/sup> \u03b22<\/sup> is symmetric<\/p>\n

    \u00a0(d) \u03b12<\/sup> \u2013 \u03b22<\/sup>= -(\u03b12<\/sup> \u2013 \u03b22<\/sup>)
    \n: \u03b12<\/sup> \u2013 \u03b22<\/sup>is symmetric<\/p>\n

    \u00a0(e) 3\u03b1 + 2\u03b2\u2260 3\u03b2 + 2\u03b1since \u03b1 \u2260 \u03b2
    \n:3\u03b1 + 2\u03b2 is not symmetric \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n(f) \u03b12<\/sup>+ \u03b22<\/sup> = \u03b22<\/sup>+\u03b12<\/sup>
    \n\t\t:\u03b12<\/sup> + \u03b22<\/sup>is symmetric
    \nIf \u03b1 and \u03b2 are the roots of 3x2<\/sup> \u2013 4x \u2013 1 = 0, find the value of:
    \n(a) \u03b1+ \u03b2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) \u03b1\u03b2
    \n(c) \u03b12<\/sup> \u03b22<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(d)
    \n(e)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(f) \u03b13<\/sup>\u03b23<\/sup>
    \n\t\t(g) \u03b1\u2013\u03b2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(h)<\/p>\n

    \u00a0Solution
    \n<\/strong>a = 3; b = -4; c = -1
    \n(a) \u03b1 + \u03b2 = \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0(b) \u03b1\u03b2 =
    \n(c) \u03b12<\/sup> \u03b22<\/sup> = (\u03b1 + \u03b2)2<\/sup> – 2\u03b1\u03b2
    \n= \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n\"\"\/(d) ==
    \n\u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0\"\"\/(e) = \u03b12<\/sup>\u03b22<\/sup> =\u00a0\u00a0\u00a0\u00a0
    \n\"\"\/\u00a0\u00a0\u00a0\u00a0\u03b1\u03b2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0(f) \u03b13<\/sup>\u03b23<\/sup> = (\u03b1+\u03b2) (\u03b12<\/sup>+\u03b22<\/sup> \u2013 \u03b1\u03b2)
    \n\u00a0\u00a0\u00a0\u00a0 = (\u03b1+\u03b2) (\u03b12<\/sup>+\u03b2)2<\/sup>-3\u03b1\u03b2)]
    \n\"\"\/\"\"\/\u00a0\u00a0\u00a0\u00a0=
    \n\u00a0\u00a0\u00a0\u00a0
    \n= <\/p>\n

    \u00a0(g) We know that
    \n(\u03b1 \u2013 \u03b2)2<\/sup> = \u03b12<\/sup>+\u03b22<\/sup> – 2\u03b1\u03b2
    \n\u00a0\u00a0\u00a0\u00a0= (\u03b1\u03b2)2<\/sup> – 4\u03b1\u03b2
    \n\"\"\/(\u03b1-\u03b2) = (\u03b1\u03b2)2<\/sup> – 4\u03b1\u03b2
    \n\"\"\/
    \n\t\t=\u00a0\u00a0\u00a0\u00a0
    \n\"\"\/\u00a0\u00a0\u00a0\u00a0
    \n=
    \n=
    \n(h) =
    \n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0=
    \n=
    \n\"\"\/\u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0=
    \n\"\"\/ 2
    \n= <\/p>\n

    \u00a0\"\"\/The Graph of y = ax2<\/sup> + bx + c (a \u2260 0) is called a parabola and has two shapes depending on whether a > 0 or a < 0.
    \n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0Q
    \n\"\"\/(a) \u00a0\u00a0\u00a0\u00a0a> 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b)\u00a0\u00a0\u00a0\u00a0
    \n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0a< 0
    \n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0P<\/p>\n

    \u00a0When a > 0, the lowest point on the graph is called the minimum point, and it occurs when
    \nx = <\/p>\n

    \t\t\"\"\/\"\"\/Also, the line when <\/p>\n

    \u00a0\u00a0\u00a0\u00a0\u00a0a> 0
    \n(a)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0a < 0
    \n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x<\/p>\n

    \u00a0x = \u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x = <\/p>\n

    \u00a0Nature of Roots
    \n<\/strong>We recall that the solution of
    \nax2<\/sup> + bc + c = 0
    \nis x = , where D = b2<\/sup> \u2013 4ac
    \nThree restrictions can be placed on the value of D.
    \n(a) D > 0
    \n(b) D < 0<\/p>\n

    \u00a0(c) D = 0
    \nWhen D > 0<\/p>\n

    \u00a0\"\"\/\"\"\/The roots of the equation are real and distinct. The graph of y = ax2<\/sup> + bx + c crosses the x \u2013 axis at two points.
    \n\"\"\/(a) \u00a0\u00a0\u00a0\u00a0a> 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) \u00a0\u00a0\u00a0\u00a0 a < 0
    \n\"\"\/\u00a0\u00a0\u00a0\u00a0 D > 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 D > 0<\/p>\n

    \u00a0x = \u03b11<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x = \u03b21<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x = \u03b12<\/sub>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x = \u03b22<\/sub>
    \n\t\tIf in addition D is a perfect square, the roots are rational, but if D is not a perfect square, the roots are irrational and are always in conjugate pairs.
    \nWhen D < 0
    \nThe roots are not real. They are said to be imaginary as is not a real number. The graph of
    \ny = ax2<\/sup> + bx + c does not cross the x \u2013 axis in this case.
    \n\"\"\/\"\"\/\"\"\/(a) \u00a0\u00a0\u00a0\u00a0a> 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0a \u2013 axis
    \n\u00a0\u00a0\u00a0\u00a0 D < 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b)\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0a < 0
    \n\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0D < 0\u00a0\u00a0\u00a0\u00a0
    \n\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0x \u2013 axis
    \nWhen D = 0
    \nThe roots are real and equal. They are said to be coincidental. The graph touches the x \u2013 axis at
    \nx = \u00a0\u00a0\u00a0\u00a0<\/p>\n

    \u00a0\"\"\/\"\"\/\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0 x =
    \n(a) \u00a0\u00a0\u00a0\u00a0a> 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0a < 0
    \n\u00a0\u00a0\u00a0\u00a0D < 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0D = 0<\/p>\n

    \u00a0x = <\/p>\n

    \u00a0Since D enables us to determine the position of the graph of y = ax2<\/sup> + bx + c relative to the x \u2013 axis, it is called a discriminant.
    \nDetermine the nature of roots of the following quadratic equations:
    \n(i) x2<\/sup> \u2013 3x \u2013 2 = 0
    \n(ii) x2<\/sup> \u2013 6x + 9 = 0
    \n(iii) 2x2<\/sup> \u2013 2x + 5 = 0<\/p>\n

    \u00a0Solution
    \n<\/strong>(i) a = 1; b = -3; c = -2
    \nD = b2<\/sup> \u2013 4ac
    \n= 9 + 8
    \n= 17<0
    \nHence the roots of the equation are real and distinct.<\/p>\n

    \u00a0(ii) x2<\/sup> \u2013 2x + 9 = 0
    \na = 1; b = -6; c = 9
    \nD = b2<\/sup> \u2013 4ac
    \n= 36 \u2013 36
    \n= 0
    \nHence the roots are real and equal.<\/p>\n

    \u00a0(iii) 2x2<\/sup> \u2013 2x + 5 = 0
    \na = 2; b = -2; c = 5
    \nD = b2<\/sup> \u2013 4ac
    \n= 4 \u2013 40
    \n= -36
    \nHence the roots are imaginary.<\/p>\n

    \u00a0Evaluation
    \n<\/strong>1. Find the quadratic equation where roots are
    \n(a) 3 and -2\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) \u00be and \u00bd <\/p>\n

    \u00a0General Evaluation
    \n<\/strong>(1) If \u03b1 and \u03b2 are the roots of 3x2<\/sup> \u2013 4x \u2013 1 = 10, find the value of:
    \n(a) \u03b12<\/sup> + \u03b22<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) (c)
    \n(d) \u03b13<\/sup> + \u03b23<\/sup>\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(e) \u03b1 \u2013 \u03b2<\/p>\n

    \u00a0(2) Find the sum and product of roots of these equation
    \n(a) 2x2<\/sup> + 3x \u2013 1 = 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) 3x2<\/sup> \u2013 5x \u2013 2 = 0<\/p>\n

    \u00a0Reading assignment
    \n<\/strong>New Further Maths Project 2 page 8, 9, 10, 11<\/p>\n

    \u00a0Weekend Assignment
    \n<\/strong>(1) Determine the nature of roots of x2<\/sup> \u2013 3x \u2013 2 = 0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0
    \n(a) Real\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) Imaginary\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(c) Equal\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(d) Coincidental
    \n(2) If \u03b1 \u2260 \u03b2 which of the following is not symmetric
    \n(a) \u03b1\u03b2 = \u03b2\u03b1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(b) \u03b1 + \u03b2 = \u03b2 + \u03b1\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0(c) 3\u03b1 + 2\u03b2 = 3\u03b2 + 2\u03b1
    \n(d) \u03b12<\/sup> + \u03b22<\/sup> = \u03b22<\/sup> + \u03b12<\/sup>
    \n\t\tIf \u03b1 and \u03b2 are the roots of 2x2<\/sup> \u2013 7x \u2013 3 = 0, find:
    \n(3)\u03b1\u03b22<\/sup> + \u03b12<\/sup>
    \n\t\t(a)
    \n(4)
    \n(a)
    \n(5)
    \n(a) <\/p>\n

    \u00a0Theory
    \n<\/strong>(1) Find the constants p, q and r such that 3x2<\/sup> \u2013 12x + 16 = p (x + q)2<\/sup> + r
    \n(2) If \u03b1 and \u03b2 are the roots of x2<\/sup> \u2013 10x + 2 = 0, find \u03b13<\/sup> \u2013 \u03b23<\/sup>.<\/p>\n

    \u00a0
    \n\u00a0
    \n\u00a0
    \n\u00a0
    \n\u00a0
    \n\u00a0
    \n\t\t\t<\/strong>\u00a0<\/p>\n","protected":false},"excerpt":{"rendered":"

    SUBJECT: FURTHER MATHEMATICS\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0\u00a0CLASS: SS2 \u00a0FIRST TERM SCHEME OF WORK \u00a0 WEEK TOPIC 1 Finding quadratic…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1,230],"tags":[],"class_list":["post-2833","post","type-post","status-publish","format-standard","hentry","category-posts","category-first-term-ss2-further-mathematics"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2833","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/comments?post=2833"}],"version-history":[{"count":1,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2833\/revisions"}],"predecessor-version":[{"id":2834,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/posts\/2833\/revisions\/2834"}],"wp:attachment":[{"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/media?parent=2833"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/categories?post=2833"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/ecolebooks.com\/nigeria\/wp-json\/wp\/v2\/tags?post=2833"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}