Solve x^2+(x+2)(1+x-7)=0

$$x ^ { 2 } + ( x + 2 ) ( 1 + x - 7 ) = 0$$

Solve for x

$$x=\sqrt{7}+1\approx 3.645751311$$

$$x=1-\sqrt{7}\approx -1.645751311$$

Steps Using the Quadratic Formula

Subtract 7 from 1 to get -6.

$$x^{2}+\left(x+2\right)\left(-6+x\right)=0$$

Use the distributive property to multiply x+2 by -6+x and combine like terms.

$$x^{2}-4x+x^{2}-12=0$$

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

$$2x^{2}-4x-12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, -4 for b, and -12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 2\left(-12\right)}}{2\times 2}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 2\left(-12\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-4\right)±\sqrt{16-8\left(-12\right)}}{2\times 2}$$

Multiply -8 times -12.

$$x=\frac{-\left(-4\right)±\sqrt{16+96}}{2\times 2}$$

Add 16 to 96.

$$x=\frac{-\left(-4\right)±\sqrt{112}}{2\times 2}$$

Take the square root of 112.

$$x=\frac{-\left(-4\right)±4\sqrt{7}}{2\times 2}$$

The opposite of -4 is 4.

$$x=\frac{4±4\sqrt{7}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{4±4\sqrt{7}}{4}$$

Now solve the equation x=\frac{4±4\sqrt{7}}{4} when ± is plus. Add 4 to 4\sqrt{7}.

$$x=\frac{4\sqrt{7}+4}{4}$$

Divide 4+4\sqrt{7} by 4.

$$x=\sqrt{7}+1$$

Now solve the equation x=\frac{4±4\sqrt{7}}{4} when ± is minus. Subtract 4\sqrt{7} from 4.

$$x=\frac{4-4\sqrt{7}}{4}$$

Divide 4-4\sqrt{7} by 4.

$$x=1-\sqrt{7}$$

The equation is now solved.

$$x=\sqrt{7}+1 x=1-\sqrt{7}$$

Steps for Completing the Square

Subtract 7 from 1 to get -6.

$$x^{2}+\left(x+2\right)\left(-6+x\right)=0$$

Use the distributive property to multiply x+2 by -6+x and combine like terms.

$$x^{2}-4x+x^{2}-12=0$$

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

$$2x^{2}-4x-12=0$$

Add 12 to both sides. Anything plus zero gives itself.

$$2x^{2}-4x=12$$

Divide both sides by 2.

$$\frac{2x^{2}-4x}{2}=\frac{12}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\left(-\frac{4}{2}\right)x=\frac{12}{2}$$

Divide -4 by 2.

$$x^{2}-2x=\frac{12}{2}$$

Divide 12 by 2.

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

Divide -2, the coefficient of the x term, by 2 to get -1. Then add the square of -1 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

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

Add 6 to 1.

$$x^{2}-2x+1=7$$

Factor x^{2}-2x+1. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-1\right)^{2}=7$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-1\right)^{2}}=\sqrt{7}$$

Simplify.

$$x-1=\sqrt{7} x-1=-\sqrt{7}$$

Add 1 to both sides of the equation.

$$x=\sqrt{7}+1 x=1-\sqrt{7}$$