Question

Check whether the following are quadratic equations

please answer it very quickly.
it's urgent
​

Answer 1

Step-by-step explanation:

please refer to the given images for solutions of 1, 2, 3 ,4 and 5

Answer 2

Answer:

(i) Yes

(ii) Yes

(iii) No

(iv) Yes

(v) Yes

(vi) No

(vii) No

(viii) Yes

Step-by-step explanation:

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

$x^{2}+2x+1=2x-6$ $($∵ $(a+b)^{2}=a^{2}+b^{2}+2ab)$

$x^{2}+2x-2x+1+6=0$

$x^{2}+7=0$

$x^{2}+7$ is a quadratic polynomial as its degree of polynomial is 2.

∴ $(x+1)^{2}=2(x-3)$ is a quadratic equation

(ii) $x^{2}-2x=(-2)(3-x)$

$x^{2}-2x=2x-6$

$x^{2}-2x-2x+6=0$

$x^{2}-4x+6=0$

$x^{2}-4x+6$ is a quadratic polynomial as its degree of polynomial is 2.

∴ $x^{2}-2x=(-2)(3-x)$ is a quadratic equation.

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

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

$($∵ $(x+a)(x+b)=x^{2}+(a+b)x+ab)$

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

$x^{2}-x^{2}-2x-x+3-2=0$

$-3x+1=0$

$3x-1$ is not a quadratic polynomial as its degree of polynomial is not 2.

∴ $(x-2)(x+1)=(x-1)(x+3)$ is not a quadratic equation.

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

$2x^{2}+x-6x-3=x^{2}+5x$

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

$2x^{2}-x^{2}-5x-5x-3=0$

$x^{2}-10x-3=0$

$x^{2}-10x-3$ is a quadratic polynomial as its degree of polynomial is 2.

∴ $(x-3)(2x+1)=x(x+5)$ is a quadratic equation.

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

$2x^{2}-6x-x+3=x^{2}+(5-1)x+(5)(-1)$

$($∵ $(x+a)(x+b)=x^{2}+(a+b)x+ab)$

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

$2x^{2}-x^{2}-7x-4x+3+5=0$

$x^{2}-11x+8=0$

$x^{2}-11x+8$ is a quadratic polynomial as its degree of polynomial is 2.

∴ $(2x-1)(x-3)=(x+5)(x-1)$ is a quadratic equation.

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

$x^{2}+3x+1=x^{2}+4-4x$ $($∵ $(a-b)^{2}=a^{2}+b^{2}-2ab)$

$x^{2}-x^{2}+3x+4x+1-4=0$

$7x-3=0$

$7x-3$ is a not quadratic polynomial as its degree of polynomial is not 2.

∴ $x^{2}+3x+1=(x-2)^{2}$ is not a quadratic equation.

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

$x^{3}+8+3(2)x^{2}+12x=2x^{3}-2x$ $($∵ $(a+b)^{3}=a^{3}+b^{3}+3a^{2}b+3ab{^2})$

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

$0=2x^{3}-x^{3}-6x^{2}-12x-2x-8$

$x^{3}-6x^{2}-12x-2x-8=0$

$x^{3}-6x^{2}-14x-8=0$

$x^{3}-6x^{2}-14x-8$ is not a quadratic polynomial as its degree of polynomial is not 2.

∴ $(x+2)^{3}=2x(x^{2}-1)$ is not a quadratic equation.

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

$x^{3}-4x^{2}-x+1=x^{3}-8-3(2)x^{2}+12x$ $($∵ $(a-b)^{3}=a^{3}-b^{3}-3a^{2}b+3ab{^2})$

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

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

$2x^{2}-13x+9=0$

$2x^{2}-13x+9$ is a quadratic polynomial as its degree of polynomial is 2.

∴ $x^{3}-4x^{2}-x+1=(x-2)^{3}$ is a quadratic equation.