Question

Check whether the following are quadratic equations :
(i) ${(x+1)}^2=2(x-3)$
(ii) $x^2-2x=(-2)(3-x)$
(iii) $(x-2)(x+1)=(x-1)(x+3)$
(iv) $(x-3)(2x+1)=x(x+5)$
(v) $(2x-1)(x-3)=(x+5)(x-1)$
(vi) $x^2+3x+1={(x-2)}^2$
(vii) ${(x+2)}^3=2x(x^2-1)$
(viii) $x^3-4x^2-x+1={(x-2)}^3$

Answer 1

Quadratic equation is a polynomial equation in which the degree is $2$.

General form of a Quadratic equation is $a{x}^2+bx+c=0,a\ne 0$

i)

${(x+1)}^2=2(x-3)$

$x^2+2x+1=2x-6$

$x^2+7=0\to$ Quadratic

ii)

$x^2-2x=-2(3-x)$

$x^2-2x=-6+2x$

$x^2+4x+6=0\to$ Quadratic

iii)

$(x-2)(x+1)=(x-1)(x+3)$

$x^2-x-2=x^2+2x-3$

$3x=1\to$ Non Quadratic

iv)

$(x-3)(2x+1)=\left(x\right)(x+5)$

${2x}^2-5x-3=x^2+5x$

$x^2+10x-3=0\to$ Quadratic

v)

$(x-3)(2x-1)=(x-1)(x+5)$

${2x}^2-7x+3=x^2+4x-5$

$x^2-11x+8=0\to$ Quadratic

vi)

$x^2+3x+1={(x-2)}^2$

$x^2+3x+1=x^2-4x+4$

$7x=3\to$ Non Quadratic

vii)

${(x+2)}^3=2x(x^2-1)$

$x^3+8+6x(x+2)=2x^3-2x$

$x^3+8+6x^2+12x=2x^3-2x$

${-x}^3+6x^2+14x+8=0\to$ Non Quadratic

viii) $x^3-4x^2-x+1={(x-2)}^3$

$x^3-4x^2-x+1=x^3+3x^2(-2)+3x(-2{)}^2+(-2{)}^3$

$x^3-4x^2-x+1=x^3-6x^2+12x-8$

$x^3-4x^2-x+1-x^3+6x^2-12x+8=0$

$2x^2-13x+9=0\to$ Quadratic