Solve x+x+xx=32

$$x+x+xx=32$$

Solve for x (complex solution)

$$x=\sqrt{33}-1\approx 4.744562647$$

$$x=-\left(\sqrt{33}+1\right)\approx -6.744562647$$

Steps Using the Quadratic Formula

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

$$x+x+x^{2}=32$$

Combine x and x to get 2x.

$$2x+x^{2}=32$$

Subtract 32 from both sides.

$$2x+x^{2}-32=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+2x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-32\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-32\right)}}{2}$$

Multiply -4 times -32.

$$x=\frac{-2±\sqrt{4+128}}{2}$$

Add 4 to 128.

$$x=\frac{-2±\sqrt{132}}{2}$$

Take the square root of 132.

$$x=\frac{-2±2\sqrt{33}}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{33}}{2} when ± is plus. Add -2 to 2\sqrt{33}.

$$x=\frac{2\sqrt{33}-2}{2}$$

Divide -2+2\sqrt{33} by 2.

$$x=\sqrt{33}-1$$

Now solve the equation x=\frac{-2±2\sqrt{33}}{2} when ± is minus. Subtract 2\sqrt{33} from -2.

$$x=\frac{-2\sqrt{33}-2}{2}$$

Divide -2-2\sqrt{33} by 2.

$$x=-\sqrt{33}-1$$

The equation is now solved.

$$x=\sqrt{33}-1 x=-\sqrt{33}-1$$

Steps for Completing the Square

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

$$x+x+x^{2}=32$$

Combine x and x to get 2x.

$$2x+x^{2}=32$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+2x=32$$

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.

$$x^{2}+2x+1^{2}=32+1^{2}$$

Square 1.

$$x^{2}+2x+1=32+1$$

Add 32 to 1.

$$x^{2}+2x+1=33$$

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}.

$$\left(x+1\right)^{2}=33$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{33}$$

Simplify.

$$x+1=\sqrt{33} x+1=-\sqrt{33}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{33}-1 x=-\sqrt{33}-1$$

Solve for x

$$x=\sqrt{33}-1\approx 4.744562647$$

$$x=-\sqrt{33}-1\approx -6.744562647$$

Steps Using the Quadratic Formula

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

$$x+x+x^{2}=32$$

Combine x and x to get 2x.

$$2x+x^{2}=32$$

Subtract 32 from both sides.

$$2x+x^{2}-32=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x^{2}+2x-32=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 2 for b, and -32 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-2±\sqrt{2^{2}-4\left(-32\right)}}{2}$$

Square 2.

$$x=\frac{-2±\sqrt{4-4\left(-32\right)}}{2}$$

Multiply -4 times -32.

$$x=\frac{-2±\sqrt{4+128}}{2}$$

Add 4 to 128.

$$x=\frac{-2±\sqrt{132}}{2}$$

Take the square root of 132.

$$x=\frac{-2±2\sqrt{33}}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{33}}{2} when ± is plus. Add -2 to 2\sqrt{33}.

$$x=\frac{2\sqrt{33}-2}{2}$$

Divide -2+2\sqrt{33} by 2.

$$x=\sqrt{33}-1$$

Now solve the equation x=\frac{-2±2\sqrt{33}}{2} when ± is minus. Subtract 2\sqrt{33} from -2.

$$x=\frac{-2\sqrt{33}-2}{2}$$

Divide -2-2\sqrt{33} by 2.

$$x=-\sqrt{33}-1$$

The equation is now solved.

$$x=\sqrt{33}-1 x=-\sqrt{33}-1$$

Steps for Completing the Square

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

$$x+x+x^{2}=32$$

Combine x and x to get 2x.

$$2x+x^{2}=32$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$x^{2}+2x=32$$

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.

$$x^{2}+2x+1^{2}=32+1^{2}$$

Square 1.

$$x^{2}+2x+1=32+1$$

Add 32 to 1.

$$x^{2}+2x+1=33$$

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}.

$$\left(x+1\right)^{2}=33$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{33}$$

Simplify.

$$x+1=\sqrt{33} x+1=-\sqrt{33}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{33}-1 x=-\sqrt{33}-1$$