Solve a^2+(a+24)^2=468

$$a ^ { 2 } + ( a + 24 ) ^ { 2 } = 468$$

Solve for a

$$a=3\sqrt{10}-12\approx -2.513167019$$

$$a=-3\sqrt{10}-12\approx -21.486832981$$

Steps Using the Quadratic Formula

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+24\right)^{2}.

$$a^{2}+a^{2}+48a+576=468$$

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

$$2a^{2}+48a+576=468$$

Subtract 468 from both sides.

$$2a^{2}+48a+576-468=0$$

Subtract 468 from 576 to get 108.

$$2a^{2}+48a+108=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 2 for a, 48 for b, and 108 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-48±\sqrt{48^{2}-4\times 2\times 108}}{2\times 2}$$

Square 48.

$$a=\frac{-48±\sqrt{2304-4\times 2\times 108}}{2\times 2}$$

Multiply -4 times 2.

$$a=\frac{-48±\sqrt{2304-8\times 108}}{2\times 2}$$

Multiply -8 times 108.

$$a=\frac{-48±\sqrt{2304-864}}{2\times 2}$$

Add 2304 to -864.

$$a=\frac{-48±\sqrt{1440}}{2\times 2}$$

Take the square root of 1440.

$$a=\frac{-48±12\sqrt{10}}{2\times 2}$$

Multiply 2 times 2.

$$a=\frac{-48±12\sqrt{10}}{4}$$

Now solve the equation a=\frac{-48±12\sqrt{10}}{4} when ± is plus. Add -48 to 12\sqrt{10}.

$$a=\frac{12\sqrt{10}-48}{4}$$

Divide -48+12\sqrt{10} by 4.

$$a=3\sqrt{10}-12$$

Now solve the equation a=\frac{-48±12\sqrt{10}}{4} when ± is minus. Subtract 12\sqrt{10} from -48.

$$a=\frac{-12\sqrt{10}-48}{4}$$

Divide -48-12\sqrt{10} by 4.

$$a=-3\sqrt{10}-12$$

The equation is now solved.

$$a=3\sqrt{10}-12 a=-3\sqrt{10}-12$$

Steps for Completing the Square

Use binomial theorem \left(a+b\right)^{2}=a^{2}+2ab+b^{2} to expand \left(a+24\right)^{2}.

$$a^{2}+a^{2}+48a+576=468$$

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

$$2a^{2}+48a+576=468$$

Subtract 576 from both sides.

$$2a^{2}+48a=468-576$$

Subtract 576 from 468 to get -108.

$$2a^{2}+48a=-108$$

Divide both sides by 2.

$$\frac{2a^{2}+48a}{2}=-\frac{108}{2}$$

Dividing by 2 undoes the multiplication by 2.

$$a^{2}+\frac{48}{2}a=-\frac{108}{2}$$

Divide 48 by 2.

$$a^{2}+24a=-\frac{108}{2}$$

Divide -108 by 2.

$$a^{2}+24a=-54$$

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.

$$a^{2}+24a+12^{2}=-54+12^{2}$$

Square 12.

$$a^{2}+24a+144=-54+144$$

Add -54 to 144.

$$a^{2}+24a+144=90$$

Factor a^{2}+24a+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}.

$$\left(a+12\right)^{2}=90$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a+12\right)^{2}}=\sqrt{90}$$

Simplify.

$$a+12=3\sqrt{10} a+12=-3\sqrt{10}$$

Subtract 12 from both sides of the equation.

$$a=3\sqrt{10}-12 a=-3\sqrt{10}-12$$