Explanation Definition used: General quadratic equation : “Given that a , b , and c are constant ( a ≠ 0 ) , the equation a x 2 + b x + c = 0 is called the general quadratic equation in x ”. Calculation: Consider the equation ( x 2 − 2 x ) 2 − 11 ( x 2 − 2 x ) + 24 = 0 . In order to find the value of x , first rewrite the equation in quadratic form. Substitute y = x 2 − 2 x in ( x 2 − 2 x ) 2 − 11 ( x 2 − 2 x ) + 24 = 0 and simplify the equation as follows. y 2 − 11 y + 24 = 0 y 2 − 8 y − 3 y + 24 = 0 y ( y − 8 ) − 3 ( y − 8 ) = 0 ( y − 8 ) ( y − 3 ) = 0 On further simplification, y − 8 = 0 or y − 3 = 0 y = 8 or y = 3 x 2 − 2 x = 8 or x 2 − 2 x = 3 [ ∵ y = x 2 − 2 x ] x 2 − 2 x − 8 = 0 or x 2 − 2 x − 3 = 0 On further simplification, ( x − 4 ) ( x + 2 ) = 0 or ( x − 3 ) ( x + 1 ) = 0 x − 4 = 0 or x + 2 = 0 or x − 3 = 0 or x + 1 = 0 x = 4 x = − 2 or x = 3 x = − 1 Thus, the solutions of the equation are x = − 1 , x = − 2 , x = 3 and x = 4 . Check the answer as follows. Substitute x = − 1 in ( x 2 − 2 x ) 2 − 11 ( x 2 − 2 x ) + 24 = 0

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Chapter 14.3, Problem 19E

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To solve: The equation (x2−2x)2−11(x2−2x)+24=0 algebraically.

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Chapter 14.3, Problem 18E

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