Solve (x-7)(x+3)+(x-1)(x+5)=102

$$( x - 7 ) ( x + 3 ) + ( x - 1 ) ( x + 5 ) = 102$$

Solve for x

$$x=-8$$

$$x=8$$

Steps by Finding Square Root

Use the distributive property to multiply x-7 by x+3 and combine like terms.

$$x^{2}-4x-21+\left(x-1\right)\left(x+5\right)=102$$

Use the distributive property to multiply x-1 by x+5 and combine like terms.

$$x^{2}-4x-21+x^{2}+4x-5=102$$

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

$$2x^{2}-4x-21+4x-5=102$$

Combine -4x and 4x to get 0.

$$2x^{2}-21-5=102$$

Subtract 5 from -21 to get -26.

$$2x^{2}-26=102$$

Add 26 to both sides.

$$2x^{2}=102+26$$

Add 102 and 26 to get 128.

$$2x^{2}=128$$

Divide both sides by 2.

$$x^{2}=\frac{128}{2}$$

Divide 128 by 2 to get 64.

$$x^{2}=64$$

Take the square root of both sides of the equation.

$$x=8 x=-8$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x-7 by x+3 and combine like terms.

$$x^{2}-4x-21+\left(x-1\right)\left(x+5\right)=102$$

Use the distributive property to multiply x-1 by x+5 and combine like terms.

$$x^{2}-4x-21+x^{2}+4x-5=102$$

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

$$2x^{2}-4x-21+4x-5=102$$

Combine -4x and 4x to get 0.

$$2x^{2}-21-5=102$$

Subtract 5 from -21 to get -26.

$$2x^{2}-26=102$$

Subtract 102 from both sides.

$$2x^{2}-26-102=0$$

Subtract 102 from -26 to get -128.

$$2x^{2}-128=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 0 for b, and -128 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{0±\sqrt{0^{2}-4\times 2\left(-128\right)}}{2\times 2}$$

Square 0.

$$x=\frac{0±\sqrt{-4\times 2\left(-128\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{0±\sqrt{-8\left(-128\right)}}{2\times 2}$$

Multiply -8 times -128.

$$x=\frac{0±\sqrt{1024}}{2\times 2}$$

Take the square root of 1024.

$$x=\frac{0±32}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{0±32}{4}$$

Now solve the equation x=\frac{0±32}{4} when ± is plus. Divide 32 by 4.

$$x=8$$

Now solve the equation x=\frac{0±32}{4} when ± is minus. Divide -32 by 4.

$$x=-8$$

The equation is now solved.

$$x=8 x=-8$$