Check whether the following are quadratic equations:(i)(x+1)^2=2(x-3)
(i) (x + 1)2 = 2(x – 3)
x2 + 2x + 1 = 2x – 6
x2 + 2x – 2x +1 + 6 = 0
x2 + 0x + 7 = 0
This is in the form of ax2 + bx + c = 0.
∴ This is a quadratic equation.
(ii) x2 – 2x = (-2) (3 – x)
x2 – 2x = -6 + 2
x2 – 2x – 2x + 6 = 0
x2 – 4x + 6 = 0
This is in the form of ax2 + bx + c = 0.
∴ This is a quadratic equation.
(iii) (x – 2) (x + 1) = (x – 1) (x + 3)
x2 + x – 2x – 2 = x2 + 3x – x – 3
x2 – x2 = x2 + 2x – 3
x2 – x2 – x – 2x – 2 + 3 = 0
-x – 2x – 2 + 3 = 0
-3x + 1 = 0
3x – 1 = 0
This is not in the form of ax2 + bx + c = 0
∴ Given equation is not quadratic equation.
(iv) (x – 3) (2x + 1) = x (x + 5)
2x2 + x – 6x – 3 = x2 + 5x
2x2 – 5x – 3 = x2 + 5x
2x2 – x2 – 5x – 5x – 3 = 0
x2 – 10x – 3 = 0
This is in the form of ax2 + bx + c = 0.
∴ This is a quadratic equation.
(v) (2x – 1) (x – 3) = (x + 5) (x – 1)
2x2 – 6x – x + 3 = x2 – x + 5x – 5
2x2 – 7x + 3 = x2 + 4x – 5
2x2 – x2 – 7x – 4x + 3 + 5 = 0
x2 – 11x + 8 = 0
This is in the form of ax2 + bx + c = 0.
∴ This is a quadratic equation.
(vi) x2 + 3x + 1 = {x – 2}2
x2 + 3x + 1 = x2 – 4x + 4
x2 + 3x + 1 – x2 + 4x – 4 = 0
7x – 3 = 0
This is not in the form of ax2 + bx + c = 0
∴ Given equation is not quadratic equation.
(vii) (x + 2)3 = 2x (x2 – 1)
x3 + 8 + 3(x)(2)(x+2) = 2x3 – 2x
x3 + 8 = 6x(x + 2) = 2x3 – 2x
x3 + 8 + 6x2 + 12x = 2x3 – 2x
x3 – 2x3 + 6x2 + 12x + 2x + 8 = 0
-x3 + 6x2 + 14x + 8 = 0
x3 – 6x2 – 14x – 8 = 0
This is not in the form of ax2 + bx + c = 0
∴ Given equation is not quadratic equation.
(viii) x3 – 4x2 – x + 1 = (x – 2)3
x3 – 4x2 – x + 1 = x3 – 8 – 3(x)(2)(x – 2)
x3 – 4x2 – x + 1 = x2 – 8 – 6x (x – 2)
x3 – 4x2 – x + 1 = x3 – 8 – 6x2 + 12x
x3 – x3 – 4x2 + 6x2 – x – 12x + 1 + 8 = 0
+2x2 – 13x + 9 = 0
This is in the form of ax2 + bx + c = 0.
∴ This is a quadratic equation.
(x+1)^2 =2(x-3)…x^2 +1 + 2×x×1 = 2x – 6x^2 +2x-2x+1+6 =0x^2+0x+7=0so ,it is a quadratic equation as a=1,b=0 nd c=7