Solve (x-5)(x+5)=2(x-1)(x+1)

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

Solve for x (complex solution)

$$x=-\sqrt{23}i\approx -0-4.795831523i$$

$$x=\sqrt{23}i\approx 4.795831523i$$

Steps by Finding Square Root

Consider \left(x-5\right)\left(x+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 5.

$$x^{2}-25=2\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2 by x-1.

$$x^{2}-25=\left(2x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 2x-2 by x+1 and combine like terms.

$$x^{2}-25=2x^{2}-2$$

Subtract 2x^{2} from both sides.

$$x^{2}-25-2x^{2}=-2$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}-25=-2$$

Add 25 to both sides.

$$-x^{2}=-2+25$$

Add -2 and 25 to get 23.

$$-x^{2}=23$$

Divide both sides by -1.

$$x^{2}=-23$$

The equation is now solved.

$$x=\sqrt{23}i x=-\sqrt{23}i$$

Steps Using the Quadratic Formula

Consider \left(x-5\right)\left(x+5\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}. Square 5.

$$x^{2}-25=2\left(x-1\right)\left(x+1\right)$$

Use the distributive property to multiply 2 by x-1.

$$x^{2}-25=\left(2x-2\right)\left(x+1\right)$$

Use the distributive property to multiply 2x-2 by x+1 and combine like terms.

$$x^{2}-25=2x^{2}-2$$

Subtract 2x^{2} from both sides.

$$x^{2}-25-2x^{2}=-2$$

Combine x^{2} and -2x^{2} to get -x^{2}.

$$-x^{2}-25=-2$$

Add 2 to both sides.

$$-x^{2}-25+2=0$$

Add -25 and 2 to get -23.

$$-x^{2}-23=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -1 for a, 0 for b, and -23 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\left(-1\right)\left(-23\right)}}{2\left(-1\right)}$$

Square 0.

$$x=\frac{0±\sqrt{-4\left(-1\right)\left(-23\right)}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$x=\frac{0±\sqrt{4\left(-23\right)}}{2\left(-1\right)}$$

Multiply 4 times -23.

$$x=\frac{0±\sqrt{-92}}{2\left(-1\right)}$$

Take the square root of -92.

$$x=\frac{0±2\sqrt{23}i}{2\left(-1\right)}$$

Multiply 2 times -1.

$$x=\frac{0±2\sqrt{23}i}{-2}$$

Now solve the equation x=\frac{0±2\sqrt{23}i}{-2} when ± is plus.

$$x=-\sqrt{23}i$$

Now solve the equation x=\frac{0±2\sqrt{23}i}{-2} when ± is minus.

$$x=\sqrt{23}i$$

The equation is now solved.

$$x=-\sqrt{23}i x=\sqrt{23}i$$