Solve (X+12)^2+(4+12)^2=35^2

$$( X + 12 ) ^ { 2 } + ( 4 + 12 ) ^ { 2 } = 35 ^ { 2 }$$

Solve for X

$$X=\sqrt{969}-12\approx 19.128764833$$

$$X=-\left(\sqrt{969}+12\right)\approx -43.128764833$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(X+12\right)^{2}.

$$X^{2}+24X+144+\left(4+12\right)^{2}=35^{2}$$

Add 4 and 12 to get 16.

$$X^{2}+24X+144+16^{2}=35^{2}$$

Calculate 16 to the power of 2 and get 256.

$$X^{2}+24X+144+256=35^{2}$$

Add 144 and 256 to get 400.

$$X^{2}+24X+400=35^{2}$$

Calculate 35 to the power of 2 and get 1225.

$$X^{2}+24X+400=1225$$

Subtract 1225 from both sides.

$$X^{2}+24X+400-1225=0$$

Subtract 1225 from 400 to get -825.

$$X^{2}+24X-825=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 24 for b, and -825 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$X=\frac{-24±\sqrt{24^{2}-4\left(-825\right)}}{2}$$

Square 24.

$$X=\frac{-24±\sqrt{576-4\left(-825\right)}}{2}$$

Multiply -4 times -825.

$$X=\frac{-24±\sqrt{576+3300}}{2}$$

Add 576 to 3300.

$$X=\frac{-24±\sqrt{3876}}{2}$$

Take the square root of 3876.

$$X=\frac{-24±2\sqrt{969}}{2}$$

Now solve the equation X=\frac{-24±2\sqrt{969}}{2} when ± is plus. Add -24 to 2\sqrt{969}.

$$X=\frac{2\sqrt{969}-24}{2}$$

Divide -24+2\sqrt{969} by 2.

$$X=\sqrt{969}-12$$

Now solve the equation X=\frac{-24±2\sqrt{969}}{2} when ± is minus. Subtract 2\sqrt{969} from -24.

$$X=\frac{-2\sqrt{969}-24}{2}$$

Divide -24-2\sqrt{969} by 2.

$$X=-\sqrt{969}-12$$

The equation is now solved.

$$X=\sqrt{969}-12 X=-\sqrt{969}-12$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(X+12\right)^{2}.

$$X^{2}+24X+144+\left(4+12\right)^{2}=35^{2}$$

Add 4 and 12 to get 16.

$$X^{2}+24X+144+16^{2}=35^{2}$$

Calculate 16 to the power of 2 and get 256.

$$X^{2}+24X+144+256=35^{2}$$

Add 144 and 256 to get 400.

$$X^{2}+24X+400=35^{2}$$

Calculate 35 to the power of 2 and get 1225.

$$X^{2}+24X+400=1225$$

Subtract 400 from both sides.

$$X^{2}+24X=1225-400$$

Subtract 400 from 1225 to get 825.

$$X^{2}+24X=825$$

Divide 24, the coefficient of the x term, by 2 to get 12. Then add the square of 12 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$X^{2}+24X+12^{2}=825+12^{2}$$

Square 12.

$$X^{2}+24X+144=825+144$$

Add 825 to 144.

$$X^{2}+24X+144=969$$

Factor X^{2}+24X+144. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(X+12\right)^{2}=969$$

Take the square root of both sides of the equation.

$$\sqrt{\left(X+12\right)^{2}}=\sqrt{969}$$

Simplify.

$$X+12=\sqrt{969} X+12=-\sqrt{969}$$

Subtract 12 from both sides of the equation.

$$X=\sqrt{969}-12 X=-\sqrt{969}-12$$