Solve (4+2x)*x=400

$$( 4 + 2 x ) \cdot x = 400$$

Solve for x (complex solution)

$$x=\sqrt{201}-1\approx 13.177446879$$

$$x=-\left(\sqrt{201}+1\right)\approx -15.177446879$$

Steps Using the Quadratic Formula

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

$$4x+2x^{2}=400$$

Subtract 400 from both sides.

$$4x+2x^{2}-400=0$$

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.

$$2x^{2}+4x-400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-400\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-400\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-400\right)}}{2\times 2}$$

Multiply -8 times -400.

$$x=\frac{-4±\sqrt{16+3200}}{2\times 2}$$

Add 16 to 3200.

$$x=\frac{-4±\sqrt{3216}}{2\times 2}$$

Take the square root of 3216.

$$x=\frac{-4±4\sqrt{201}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±4\sqrt{201}}{4}$$

Now solve the equation x=\frac{-4±4\sqrt{201}}{4} when ± is plus. Add -4 to 4\sqrt{201}.

$$x=\frac{4\sqrt{201}-4}{4}$$

Divide -4+4\sqrt{201} by 4.

$$x=\sqrt{201}-1$$

Now solve the equation x=\frac{-4±4\sqrt{201}}{4} when ± is minus. Subtract 4\sqrt{201} from -4.

$$x=\frac{-4\sqrt{201}-4}{4}$$

Divide -4-4\sqrt{201} by 4.

$$x=-\sqrt{201}-1$$

The equation is now solved.

$$x=\sqrt{201}-1 x=-\sqrt{201}-1$$

Steps for Completing the Square

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

$$4x+2x^{2}=400$$

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.

$$2x^{2}+4x=400$$

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=\frac{400}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=\frac{400}{2}$$

Divide 4 by 2.

$$x^{2}+2x=\frac{400}{2}$$

Divide 400 by 2.

$$x^{2}+2x=200$$

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.

$$x^{2}+2x+1^{2}=200+1^{2}$$

Square 1.

$$x^{2}+2x+1=200+1$$

Add 200 to 1.

$$x^{2}+2x+1=201$$

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}.

$$\left(x+1\right)^{2}=201$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{201}$$

Simplify.

$$x+1=\sqrt{201} x+1=-\sqrt{201}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{201}-1 x=-\sqrt{201}-1$$

Solve for x

$$x=\sqrt{201}-1\approx 13.177446879$$

$$x=-\sqrt{201}-1\approx -15.177446879$$

Steps Using the Quadratic Formula

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

$$4x+2x^{2}=400$$

Subtract 400 from both sides.

$$4x+2x^{2}-400=0$$

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.

$$2x^{2}+4x-400=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 4 for b, and -400 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-400\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-400\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-400\right)}}{2\times 2}$$

Multiply -8 times -400.

$$x=\frac{-4±\sqrt{16+3200}}{2\times 2}$$

Add 16 to 3200.

$$x=\frac{-4±\sqrt{3216}}{2\times 2}$$

Take the square root of 3216.

$$x=\frac{-4±4\sqrt{201}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±4\sqrt{201}}{4}$$

Now solve the equation x=\frac{-4±4\sqrt{201}}{4} when ± is plus. Add -4 to 4\sqrt{201}.

$$x=\frac{4\sqrt{201}-4}{4}$$

Divide -4+4\sqrt{201} by 4.

$$x=\sqrt{201}-1$$

Now solve the equation x=\frac{-4±4\sqrt{201}}{4} when ± is minus. Subtract 4\sqrt{201} from -4.

$$x=\frac{-4\sqrt{201}-4}{4}$$

Divide -4-4\sqrt{201} by 4.

$$x=-\sqrt{201}-1$$

The equation is now solved.

$$x=\sqrt{201}-1 x=-\sqrt{201}-1$$

Steps for Completing the Square

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

$$4x+2x^{2}=400$$

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.

$$2x^{2}+4x=400$$

Divide both sides by 2.

$$\frac{2x^{2}+4x}{2}=\frac{400}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{4}{2}x=\frac{400}{2}$$

Divide 4 by 2.

$$x^{2}+2x=\frac{400}{2}$$

Divide 400 by 2.

$$x^{2}+2x=200$$

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.

$$x^{2}+2x+1^{2}=200+1^{2}$$

Square 1.

$$x^{2}+2x+1=200+1$$

Add 200 to 1.

$$x^{2}+2x+1=201$$

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}.

$$\left(x+1\right)^{2}=201$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{201}$$

Simplify.

$$x+1=\sqrt{201} x+1=-\sqrt{201}$$

Subtract 1 from both sides of the equation.

$$x=\sqrt{201}-1 x=-\sqrt{201}-1$$