Mathematics
Find which of the following equations are quadratic :
(i) (3x - 1)2 = 5(x + 8)
(ii) 5x2 - 8x = -3(7 - 2x)
(iii) (x - 4)(3x + 1) = (3x - 1)(x + 2)
(iv) x2 + 5x - 5 = (x - 3)2
(v) 7x3 - 2x2 + 10 = (2x - 5)2
(vi) (x - 1)2 + (x + 2)2 + 3(x + 1) = 0
Quadratic Equations
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Answer
(i) (3x - 1)2 = 5(x + 8)
⇒ 9x2 + 1 - 6x = 5x + 40
⇒ 9x2 - 6x - 5x + 1 - 40 = 0
⇒ 9x2 - 11x - 39 = 0 which is of the form ax2 + bx + c = 0.
∴ Given equation is a quadratic equation.
(ii) 5x2 - 8x = -3(7 - 2x)
⇒ 5x2 - 8x = -21 + 6x
⇒ 5x2 - 8x - 6x + 21 = 0
⇒ 5x2 - 14x + 21 = 0 which is of the form ax2 + bx + c = 0.
∴ Given equation is a quadratic equation.
(iii) (x - 4)(3x + 1) = (3x - 1)(x + 2)
⇒ 3x2 + x - 12x - 4 = 3x2 + 6x - x - 2
⇒ 3x2 - 3x2 - 11x - 5x - 4 + 2 = 0
⇒ -16x - 2 = 0
⇒ 16x + 2 = 0 which is not of the form ax2 + bx + c = 0.
∴ Given equation is not a quadratic equation.
(iv) x2 + 5x - 5 = (x - 3)2
⇒ x2 + 5x - 5 = x2 + 9 - 6x
⇒ x2 - x2 + 5x + 6x - 5 - 9 = 0
11x - 14 = 0 which is not of the form ax2 + bx + c = 0.
∴ Given equation is not a quadratic equation.
(v) 7x3 - 2x2 + 10 = (2x - 5)2
⇒ 7x3 - 2x2 + 10 = 4x2 + 25 - 20x
⇒ 7x3 - 2x2 - 4x2 + 10 - 25 + 20x = 0
⇒ 7x3 - 6x2 - 15 + 20x = 0 which is not of the form ax2 + bx + c = 0.
∴ Given equation is not a quadratic equation.
(vi) (x - 1)2 + (x + 2)2 + 3(x + 1) = 0
⇒ x2 + 1 - 2x + x2 + 4 + 4x + 3x + 3 = 0
⇒ 2x2 + 5x + 8 = 0 which is of the form ax2 + bx + c = 0.
∴ Given equation is a quadratic equation.
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