Check whether the following are quadratic equation?

An equation of the form ax²+bx+c = 0, a ≠ 0 and a, b, c are real numbers is called quadratic equation.

(i) (x+1)² = 2(x-3)
x²+2x+1 = 2x-6
x²+2x+1 -2x+6 = 0
x²+7 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 1≠ 0, b = 0, c = 7
Hence the given equation is quadratic equation.

(ii) x²-2x = (-2)(3-x)
x²-2x  = -6+2x
x²-2x -2x+6 = 0
x²-4x+6 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 1≠ 0, b = -4, c = 6
Hence the given equation is quadratic equation.

(iii) (x-2)(x+1) = (x-1)(x+3)
x²-2x+x-2 = x²-x+3x-3
x²-2x+x-2 -x²+x-3x+3= 0
-3x+1 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 0, b = -3, c = 1
Hence the given equation is not quadratic equation.

(iv) (x-3)(2x+1) = x(x+5)
x²-6x+x-3 = x²+5x
x²-6x+x-3 -x²-5x = 0
-10x-3 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 0, b = -10, c = -3
Hence the given equation is not quadratic equation.

(v) (2x-1)(x-3) = (x+5)(x-1)
2x²-6x-x+3 = x²+5x-x-5
2x²-6x-x+3 -x²-5x+x+5 = 0
x²-11x+8 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 1 ≠ 0, b = -11, c = 8
Hence the given equation is 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
Comparing above equation with ax²+bx+c = 0, we have a = 0, b = 7, c = -3
Hence the given equation is not quadratic equation.

(vii) (x+2)³ = 2x(x²-1)
x³+6x²+12x+8 = 2x³-2x
x³+6x²+12x+8 -2x³+2x = 0
x³ – 6x²-14x-8 = 0
It is not in the form of ax²+bx+c = 0s
Hence the given equation is not a quadratic equation.

(viii) x³-4x²-x+1 = (x-2)³
x³-4x²-x+1 = x³-6x²+12x-8
x³-4x²-x+1 -x³+6x²-12x+8= 0
2x²-13x+9 = 0
Comparing above equation with ax²+bx+c = 0, we have a = 2 ≠ 0, b = -13, c = 9
Hence the given equation is quadratic equation.