Solve r^2-23r+24=0

$${ r }^{ 2 } -23r+24=0$$

Solve for r

$$r = \frac{\sqrt{433} + 23}{2} \approx 21.904326023$$

$$r = \frac{23 - \sqrt{433}}{2} \approx 1.095673977$$

Steps Using the Quadratic Formula

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.

$$r^{2}-23r+24=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -23 for b, and 24 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$r=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 24}}{2}$$

Square -23.

$$r=\frac{-\left(-23\right)±\sqrt{529-4\times 24}}{2}$$

Multiply -4 times 24.

$$r=\frac{-\left(-23\right)±\sqrt{529-96}}{2}$$

Add 529 to -96.

$$r=\frac{-\left(-23\right)±\sqrt{433}}{2}$$

The opposite of -23 is 23.

$$r=\frac{23±\sqrt{433}}{2}$$

Now solve the equation r=\frac{23±\sqrt{433}}{2} when ± is plus. Add 23 to \sqrt{433}.

$$r=\frac{\sqrt{433}+23}{2}$$

Now solve the equation r=\frac{23±\sqrt{433}}{2} when ± is minus. Subtract \sqrt{433} from 23.

$$r=\frac{23-\sqrt{433}}{2}$$

The equation is now solved.

$$r=\frac{\sqrt{433}+23}{2} r=\frac{23-\sqrt{433}}{2}$$

Steps for Completing the Square

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.

$$r^{2}-23r+24=0$$

Subtract 24 from both sides of the equation.

$$r^{2}-23r+24-24=-24$$

Subtracting 24 from itself leaves 0.

$$r^{2}-23r=-24$$

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.

$$r^{2}-23r+\left(-\frac{23}{2}\right)^{2}=-24+\left(-\frac{23}{2}\right)^{2}$$

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

$$r^{2}-23r+\frac{529}{4}=-24+\frac{529}{4}$$

Add -24 to \frac{529}{4}.

$$r^{2}-23r+\frac{529}{4}=\frac{433}{4}$$

Factor r^{2}-23r+\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}.

$$\left(r-\frac{23}{2}\right)^{2}=\frac{433}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(r-\frac{23}{2}\right)^{2}}=\sqrt{\frac{433}{4}}$$

Simplify.

$$r-\frac{23}{2}=\frac{\sqrt{433}}{2} r-\frac{23}{2}=-\frac{\sqrt{433}}{2}$$

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

$$r=\frac{\sqrt{433}+23}{2} r=\frac{23-\sqrt{433}}{2}$$