Solve a^2=23a+112

$$a ^ { 2 } = 23 a + 112$$

Solve for a

$$a = \frac{\sqrt{977} + 23}{2} \approx 27.128499608$$

$$a=\frac{23-\sqrt{977}}{2}\approx -4.128499608$$

Steps Using the Quadratic Formula

Subtract 23a from both sides.

$$a^{2}-23a=112$$

Subtract 112 from both sides.

$$a^{2}-23a-112=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -23 for b, and -112 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\left(-112\right)}}{2}$$

Square -23.

$$a=\frac{-\left(-23\right)±\sqrt{529-4\left(-112\right)}}{2}$$

Multiply -4 times -112.

$$a=\frac{-\left(-23\right)±\sqrt{529+448}}{2}$$

Add 529 to 448.

$$a=\frac{-\left(-23\right)±\sqrt{977}}{2}$$

The opposite of -23 is 23.

$$a=\frac{23±\sqrt{977}}{2}$$

Now solve the equation a=\frac{23±\sqrt{977}}{2} when ± is plus. Add 23 to \sqrt{977}.

$$a=\frac{\sqrt{977}+23}{2}$$

Now solve the equation a=\frac{23±\sqrt{977}}{2} when ± is minus. Subtract \sqrt{977} from 23.

$$a=\frac{23-\sqrt{977}}{2}$$

The equation is now solved.

$$a=\frac{\sqrt{977}+23}{2} a=\frac{23-\sqrt{977}}{2}$$

Steps for Completing the Square

Subtract 23a from both sides.

$$a^{2}-23a=112$$

Divide -23, the coefficient of the x term, by 2 to get -\frac{23}{2}. Then add the square of -\frac{23}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-23a+\left(-\frac{23}{2}\right)^{2}=112+\left(-\frac{23}{2}\right)^{2}$$

Square -\frac{23}{2} by squaring both the numerator and the denominator of the fraction.

$$a^{2}-23a+\frac{529}{4}=112+\frac{529}{4}$$

Add 112 to \frac{529}{4}.

$$a^{2}-23a+\frac{529}{4}=\frac{977}{4}$$

Factor a^{2}-23a+\frac{529}{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}.

$$\left(a-\frac{23}{2}\right)^{2}=\frac{977}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{23}{2}\right)^{2}}=\sqrt{\frac{977}{4}}$$

Simplify.

$$a-\frac{23}{2}=\frac{\sqrt{977}}{2} a-\frac{23}{2}=-\frac{\sqrt{977}}{2}$$

Add \frac{23}{2} to both sides of the equation.

$$a=\frac{\sqrt{977}+23}{2} a=\frac{23-\sqrt{977}}{2}$$