Solve left(8-xright)^2+7^2=left(15-2xright)^2

$${ \left(8-x \right) }^{ 2 } + { 7 }^{ 2 } = { \left(15-2x \right) }^{ 2 }$$

Solve for x

$$x = \frac{2 \sqrt{37} + 22}{3} \approx 11.388508354$$

$$x = \frac{22 - 2 \sqrt{37}}{3} \approx 3.278158313$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-x\right)^{2}.

$$64-16x+x^{2}+7^{2}=\left(15-2x\right)^{2}$$

Calculate 7 to the power of 2 and get 49.

$$64-16x+x^{2}+49=\left(15-2x\right)^{2}$$

Add 64 and 49 to get 113.

$$113-16x+x^{2}=\left(15-2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(15-2x\right)^{2}.

$$113-16x+x^{2}=225-60x+4x^{2}$$

Subtract 225 from both sides.

$$113-16x+x^{2}-225=-60x+4x^{2}$$

Subtract 225 from 113 to get -112.

$$-112-16x+x^{2}=-60x+4x^{2}$$

Add 60x to both sides.

$$-112-16x+x^{2}+60x=4x^{2}$$

Combine -16x and 60x to get 44x.

$$-112+44x+x^{2}=4x^{2}$$

Subtract 4x^{2} from both sides.

$$-112+44x+x^{2}-4x^{2}=0$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$-112+44x-3x^{2}=0$$

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.

$$-3x^{2}+44x-112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -3 for a, 44 for b, and -112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-44±\sqrt{44^{2}-4\left(-3\right)\left(-112\right)}}{2\left(-3\right)}$$

Square 44.

$$x=\frac{-44±\sqrt{1936-4\left(-3\right)\left(-112\right)}}{2\left(-3\right)}$$

Multiply -4 times -3.

$$x=\frac{-44±\sqrt{1936+12\left(-112\right)}}{2\left(-3\right)}$$

Multiply 12 times -112.

$$x=\frac{-44±\sqrt{1936-1344}}{2\left(-3\right)}$$

Add 1936 to -1344.

$$x=\frac{-44±\sqrt{592}}{2\left(-3\right)}$$

Take the square root of 592.

$$x=\frac{-44±4\sqrt{37}}{2\left(-3\right)}$$

Multiply 2 times -3.

$$x=\frac{-44±4\sqrt{37}}{-6}$$

Now solve the equation x=\frac{-44±4\sqrt{37}}{-6} when ± is plus. Add -44 to 4\sqrt{37}.

$$x=\frac{4\sqrt{37}-44}{-6}$$

Divide -44+4\sqrt{37} by -6.

$$x=\frac{22-2\sqrt{37}}{3}$$

Now solve the equation x=\frac{-44±4\sqrt{37}}{-6} when ± is minus. Subtract 4\sqrt{37} from -44.

$$x=\frac{-4\sqrt{37}-44}{-6}$$

Divide -44-4\sqrt{37} by -6.

$$x=\frac{2\sqrt{37}+22}{3}$$

The equation is now solved.

$$x=\frac{22-2\sqrt{37}}{3} x=\frac{2\sqrt{37}+22}{3}$$

Steps for Completing the Square

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(8-x\right)^{2}.

$$64-16x+x^{2}+7^{2}=\left(15-2x\right)^{2}$$

Calculate 7 to the power of 2 and get 49.

$$64-16x+x^{2}+49=\left(15-2x\right)^{2}$$

Add 64 and 49 to get 113.

$$113-16x+x^{2}=\left(15-2x\right)^{2}$$

Use binomial theorem \left(a-b\right)^{2}=a^{2}-2ab+b^{2} to expand \left(15-2x\right)^{2}.

$$113-16x+x^{2}=225-60x+4x^{2}$$

Add 60x to both sides.

$$113-16x+x^{2}+60x=225+4x^{2}$$

Combine -16x and 60x to get 44x.

$$113+44x+x^{2}=225+4x^{2}$$

Subtract 4x^{2} from both sides.

$$113+44x+x^{2}-4x^{2}=225$$

Combine x^{2} and -4x^{2} to get -3x^{2}.

$$113+44x-3x^{2}=225$$

Subtract 113 from both sides.

$$44x-3x^{2}=225-113$$

Subtract 113 from 225 to get 112.

$$44x-3x^{2}=112$$

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.

$$-3x^{2}+44x=112$$

Divide both sides by -3.

$$\frac{-3x^{2}+44x}{-3}=\frac{112}{-3}$$

Dividing by -3 undoes the multiplication by -3.

$$x^{2}+\frac{44}{-3}x=\frac{112}{-3}$$

Divide 44 by -3.

$$x^{2}-\frac{44}{3}x=\frac{112}{-3}$$

Divide 112 by -3.

$$x^{2}-\frac{44}{3}x=-\frac{112}{3}$$

Divide -\frac{44}{3}, the coefficient of the x term, by 2 to get -\frac{22}{3}. Then add the square of -\frac{22}{3} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{44}{3}x+\left(-\frac{22}{3}\right)^{2}=-\frac{112}{3}+\left(-\frac{22}{3}\right)^{2}$$

Square -\frac{22}{3} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{44}{3}x+\frac{484}{9}=-\frac{112}{3}+\frac{484}{9}$$

Add -\frac{112}{3} to \frac{484}{9} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{44}{3}x+\frac{484}{9}=\frac{148}{9}$$

Factor x^{2}-\frac{44}{3}x+\frac{484}{9}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{22}{3}\right)^{2}=\frac{148}{9}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{22}{3}\right)^{2}}=\sqrt{\frac{148}{9}}$$

Simplify.

$$x-\frac{22}{3}=\frac{2\sqrt{37}}{3} x-\frac{22}{3}=-\frac{2\sqrt{37}}{3}$$

Add \frac{22}{3} to both sides of the equation.

$$x=\frac{2\sqrt{37}+22}{3} x=\frac{22-2\sqrt{37}}{3}$$