Solve (x(x+1))+(x(x+1))=25

$$(x(x+1))+(x(x+1))=25$$

Solve for x

$$x = \frac{\sqrt{51} - 1}{2} \approx 3.070714214$$

$$x=\frac{-\sqrt{51}-1}{2}\approx -4.070714214$$

Steps Using the Quadratic Formula

Use the distributive property to multiply x by x+1.

$$x^{2}+x+x\left(x+1\right)=25$$

Use the distributive property to multiply x by x+1.

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

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

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

Combine x and x to get 2x.

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

Subtract 25 from both sides.

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

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

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

Square 2.

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

Multiply -4 times 2.

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

Multiply -8 times -25.

$$x=\frac{-2±\sqrt{4+200}}{2\times 2}$$

Add 4 to 200.

$$x=\frac{-2±\sqrt{204}}{2\times 2}$$

Take the square root of 204.

$$x=\frac{-2±2\sqrt{51}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-2±2\sqrt{51}}{4}$$

Now solve the equation x=\frac{-2±2\sqrt{51}}{4} when ± is plus. Add -2 to 2\sqrt{51}.

$$x=\frac{2\sqrt{51}-2}{4}$$

Divide -2+2\sqrt{51} by 4.

$$x=\frac{\sqrt{51}-1}{2}$$

Now solve the equation x=\frac{-2±2\sqrt{51}}{4} when ± is minus. Subtract 2\sqrt{51} from -2.

$$x=\frac{-2\sqrt{51}-2}{4}$$

Divide -2-2\sqrt{51} by 4.

$$x=\frac{-\sqrt{51}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{51}-1}{2} x=\frac{-\sqrt{51}-1}{2}$$

Steps for Completing the Square

Use the distributive property to multiply x by x+1.

$$x^{2}+x+x\left(x+1\right)=25$$

Use the distributive property to multiply x by x+1.

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

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

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

Combine x and x to get 2x.

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

Divide both sides by 2.

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

Dividing by 2 undoes the multiplication by 2.

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

Divide 2 by 2.

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

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

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{25}{2}+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

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

Add \frac{25}{2} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{51}{4}$$

Factor x^{2}+x+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{51}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{51}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{51}}{2} x+\frac{1}{2}=-\frac{\sqrt{51}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{51}-1}{2} x=\frac{-\sqrt{51}-1}{2}$$