Solve 24=x^2+x

$$24= { x }^{ 2 } +x$$

Solve for x

$$x = \frac{\sqrt{97} - 1}{2} \approx 4.424428901$$

$$x=\frac{-\sqrt{97}-1}{2}\approx -5.424428901$$

Steps Using the Quadratic Formula

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+x=24$$

Subtract 24 from both sides.

$$x^{2}+x-24=0$$

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

$$x=\frac{-1±\sqrt{1^{2}-4\left(-24\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$x=\frac{-1±\sqrt{1+96}}{2}$$

Add 1 to 96.

$$x=\frac{-1±\sqrt{97}}{2}$$

Now solve the equation x=\frac{-1±\sqrt{97}}{2} when ± is plus. Add -1 to \sqrt{97}.

$$x=\frac{\sqrt{97}-1}{2}$$

Now solve the equation x=\frac{-1±\sqrt{97}}{2} when ± is minus. Subtract \sqrt{97} from -1.

$$x=\frac{-\sqrt{97}-1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{97}-1}{2} x=\frac{-\sqrt{97}-1}{2}$$

Steps for Completing the Square

Swap sides so that all variable terms are on the left hand side.

$$x^{2}+x=24$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=24+\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+x+\frac{1}{4}=24+\frac{1}{4}$$

Add 24 to \frac{1}{4}.

$$x^{2}+x+\frac{1}{4}=\frac{97}{4}$$

Factor x^{2}+x+\frac{1}{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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{97}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{97}{4}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{97}}{2} x+\frac{1}{2}=-\frac{\sqrt{97}}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{97}-1}{2} x=\frac{-\sqrt{97}-1}{2}$$