Solve 4x(x+2)=25

$$4 x ( x + 2 ) = 25$$

Solve for x

$$x=\frac{\sqrt{29}}{2}-1\approx 1.692582404$$

$$x=-\frac{\sqrt{29}}{2}-1\approx -3.692582404$$

Steps Using the Quadratic Formula

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

$$4x^{2}+8x=25$$

Subtract 25 from both sides.

$$4x^{2}+8x-25=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, 8 for b, and -25 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-8±\sqrt{8^{2}-4\times 4\left(-25\right)}}{2\times 4}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 4\left(-25\right)}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-8±\sqrt{64-16\left(-25\right)}}{2\times 4}$$

Multiply -16 times -25.

$$x=\frac{-8±\sqrt{64+400}}{2\times 4}$$

Add 64 to 400.

$$x=\frac{-8±\sqrt{464}}{2\times 4}$$

Take the square root of 464.

$$x=\frac{-8±4\sqrt{29}}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-8±4\sqrt{29}}{8}$$

Now solve the equation x=\frac{-8±4\sqrt{29}}{8} when ± is plus. Add -8 to 4\sqrt{29}.

$$x=\frac{4\sqrt{29}-8}{8}$$

Divide -8+4\sqrt{29} by 8.

$$x=\frac{\sqrt{29}}{2}-1$$

Now solve the equation x=\frac{-8±4\sqrt{29}}{8} when ± is minus. Subtract 4\sqrt{29} from -8.

$$x=\frac{-4\sqrt{29}-8}{8}$$

Divide -8-4\sqrt{29} by 8.

$$x=-\frac{\sqrt{29}}{2}-1$$

The equation is now solved.

$$x=\frac{\sqrt{29}}{2}-1 x=-\frac{\sqrt{29}}{2}-1$$

Steps for Completing the Square

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

$$4x^{2}+8x=25$$

Divide both sides by 4.

$$\frac{4x^{2}+8x}{4}=\frac{25}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\frac{8}{4}x=\frac{25}{4}$$

Divide 8 by 4.

$$x^{2}+2x=\frac{25}{4}$$

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.

$$x^{2}+2x+1^{2}=\frac{25}{4}+1^{2}$$

Square 1.

$$x^{2}+2x+1=\frac{25}{4}+1$$

Add \frac{25}{4} to 1.

$$x^{2}+2x+1=\frac{29}{4}$$

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}.

$$\left(x+1\right)^{2}=\frac{29}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+1\right)^{2}}=\sqrt{\frac{29}{4}}$$

Simplify.

$$x+1=\frac{\sqrt{29}}{2} x+1=-\frac{\sqrt{29}}{2}$$

Subtract 1 from both sides of the equation.

$$x=\frac{\sqrt{29}}{2}-1 x=-\frac{\sqrt{29}}{2}-1$$