Solve (x+4+x)(x)=100

$$(x+4+x)(x)=100$$

Solve for x (complex solution)

$$x=\sqrt{51}-1\approx 6.141428429$$

$$x=-\left(\sqrt{51}+1\right)\approx -8.141428429$$

Steps Using the Quadratic Formula

Combine x and x to get 2x.

$$\left(2x+4\right)x=100$$

Use the distributive property to multiply 2x+4 by x.

$$2x^{2}+4x=100$$

Subtract 100 from both sides.

$$2x^{2}+4x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-100\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-100\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-100\right)}}{2\times 2}$$

Multiply -8 times -100.

$$x=\frac{-4±\sqrt{16+800}}{2\times 2}$$

Add 16 to 800.

$$x=\frac{-4±\sqrt{816}}{2\times 2}$$

Take the square root of 816.

$$x=\frac{-4±4\sqrt{51}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±4\sqrt{51}}{4}$$

Now solve the equation x=\frac{-4±4\sqrt{51}}{4} when ± is plus. Add -4 to 4\sqrt{51}.

$$x=\frac{4\sqrt{51}-4}{4}$$

Divide -4+4\sqrt{51} by 4.

$$x=\sqrt{51}-1$$

Now solve the equation x=\frac{-4±4\sqrt{51}}{4} when ± is minus. Subtract 4\sqrt{51} from -4.

$$x=\frac{-4\sqrt{51}-4}{4}$$

Divide -4-4\sqrt{51} by 4.

$$x=-\sqrt{51}-1$$

The equation is now solved.

$$x=\sqrt{51}-1 x=-\sqrt{51}-1$$

Steps for Completing the Square

Combine x and x to get 2x.

$$\left(2x+4\right)x=100$$

Use the distributive property to multiply 2x+4 by x.

$$2x^{2}+4x=100$$

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=\frac{100}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=\frac{100}{2}$$

Divide 4 by 2.

$$x^{2}+2x=\frac{100}{2}$$

Divide 100 by 2.

$$x^{2}+2x=50$$

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.

$$x^{2}+2x+1^{2}=50+1^{2}$$

Square 1.

$$x^{2}+2x+1=50+1$$

Add 50 to 1.

$$x^{2}+2x+1=51$$

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}.

$$\left(x+1\right)^{2}=51$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{51}$$

Simplify.

$$x+1=\sqrt{51} x+1=-\sqrt{51}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{51}-1 x=-\sqrt{51}-1$$

Solve for x

$$x=\sqrt{51}-1\approx 6.141428429$$

$$x=-\sqrt{51}-1\approx -8.141428429$$

Steps Using the Quadratic Formula

Combine x and x to get 2x.

$$\left(2x+4\right)x=100$$

Use the distributive property to multiply 2x+4 by x.

$$2x^{2}+4x=100$$

Subtract 100 from both sides.

$$2x^{2}+4x-100=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -100 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-100\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-100\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-100\right)}}{2\times 2}$$

Multiply -8 times -100.

$$x=\frac{-4±\sqrt{16+800}}{2\times 2}$$

Add 16 to 800.

$$x=\frac{-4±\sqrt{816}}{2\times 2}$$

Take the square root of 816.

$$x=\frac{-4±4\sqrt{51}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±4\sqrt{51}}{4}$$

Now solve the equation x=\frac{-4±4\sqrt{51}}{4} when ± is plus. Add -4 to 4\sqrt{51}.

$$x=\frac{4\sqrt{51}-4}{4}$$

Divide -4+4\sqrt{51} by 4.

$$x=\sqrt{51}-1$$

Now solve the equation x=\frac{-4±4\sqrt{51}}{4} when ± is minus. Subtract 4\sqrt{51} from -4.

$$x=\frac{-4\sqrt{51}-4}{4}$$

Divide -4-4\sqrt{51} by 4.

$$x=-\sqrt{51}-1$$

The equation is now solved.

$$x=\sqrt{51}-1 x=-\sqrt{51}-1$$

Steps for Completing the Square

Combine x and x to get 2x.

$$\left(2x+4\right)x=100$$

Use the distributive property to multiply 2x+4 by x.

$$2x^{2}+4x=100$$

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=\frac{100}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=\frac{100}{2}$$

Divide 4 by 2.

$$x^{2}+2x=\frac{100}{2}$$

Divide 100 by 2.

$$x^{2}+2x=50$$

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.

$$x^{2}+2x+1^{2}=50+1^{2}$$

Square 1.

$$x^{2}+2x+1=50+1$$

Add 50 to 1.

$$x^{2}+2x+1=51$$

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}.

$$\left(x+1\right)^{2}=51$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{51}$$

Simplify.

$$x+1=\sqrt{51} x+1=-\sqrt{51}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{51}-1 x=-\sqrt{51}-1$$