Solve x^2+(x+7)^2=15^2

$$x ^ { 2 } + ( x + 7 ) ^ { 2 } = 15 ^ { 2 }$$

Solve for x

$$x = \frac{\sqrt{401} - 7}{2} \approx 6.512492197$$

$$x=\frac{-\sqrt{401}-7}{2}\approx -13.512492197$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+x^{2}+14x+49=15^{2}$$

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

$$2x^{2}+14x+49=15^{2}$$

Calculate 15 to the power of 2 and get 225.

$$2x^{2}+14x+49=225$$

Subtract 225 from both sides.

$$2x^{2}+14x+49-225=0$$

Subtract 225 from 49 to get -176.

$$2x^{2}+14x-176=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 14 for b, and -176 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-14±\sqrt{14^{2}-4\times 2\left(-176\right)}}{2\times 2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 2\left(-176\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-14±\sqrt{196-8\left(-176\right)}}{2\times 2}$$

Multiply -8 times -176.

$$x=\frac{-14±\sqrt{196+1408}}{2\times 2}$$

Add 196 to 1408.

$$x=\frac{-14±\sqrt{1604}}{2\times 2}$$

Take the square root of 1604.

$$x=\frac{-14±2\sqrt{401}}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-14±2\sqrt{401}}{4}$$

Now solve the equation x=\frac{-14±2\sqrt{401}}{4} when ± is plus. Add -14 to 2\sqrt{401}.

$$x=\frac{2\sqrt{401}-14}{4}$$

Divide -14+2\sqrt{401} by 4.

$$x=\frac{\sqrt{401}-7}{2}$$

Now solve the equation x=\frac{-14±2\sqrt{401}}{4} when ± is minus. Subtract 2\sqrt{401} from -14.

$$x=\frac{-2\sqrt{401}-14}{4}$$

Divide -14-2\sqrt{401} by 4.

$$x=\frac{-\sqrt{401}-7}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{401}-7}{2} x=\frac{-\sqrt{401}-7}{2}$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(x+7\right)^{2}.

$$x^{2}+x^{2}+14x+49=15^{2}$$

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

$$2x^{2}+14x+49=15^{2}$$

Calculate 15 to the power of 2 and get 225.

$$2x^{2}+14x+49=225$$

Subtract 49 from both sides.

$$2x^{2}+14x=225-49$$

Subtract 49 from 225 to get 176.

$$2x^{2}+14x=176$$

Divide both sides by 2.

$$\frac{2x^{2}+14x}{2}=\frac{176}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$x^{2}+\frac{14}{2}x=\frac{176}{2}$$

Divide 14 by 2.

$$x^{2}+7x=\frac{176}{2}$$

Divide 176 by 2.

$$x^{2}+7x=88$$

Divide 7, the coefficient of the x term, by 2 to get \frac{7}{2}. Then add the square of \frac{7}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+7x+\left(\frac{7}{2}\right)^{2}=88+\left(\frac{7}{2}\right)^{2}$$

Square \frac{7}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+7x+\frac{49}{4}=88+\frac{49}{4}$$

Add 88 to \frac{49}{4}.

$$x^{2}+7x+\frac{49}{4}=\frac{401}{4}$$

Factor x^{2}+7x+\frac{49}{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}.

$$\left(x+\frac{7}{2}\right)^{2}=\frac{401}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{7}{2}\right)^{2}}=\sqrt{\frac{401}{4}}$$

Simplify.

$$x+\frac{7}{2}=\frac{\sqrt{401}}{2} x+\frac{7}{2}=-\frac{\sqrt{401}}{2}$$

Subtract \frac{7}{2} from both sides of the equation.

$$x=\frac{\sqrt{401}-7}{2} x=\frac{-\sqrt{401}-7}{2}$$