Solve 24x^2+5x=360

$$24 { x }^{ 2 } +5x=360$$

Solve for x

$$x = \frac{\sqrt{34585} - 5}{48} \approx 3.770217245$$

$$x=\frac{-\sqrt{34585}-5}{48}\approx -3.978550578$$

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.

$$24x^{2}+5x=360$$

Subtract 360 from both sides of the equation.

$$24x^{2}+5x-360=360-360$$

Subtracting 360 from itself leaves 0.

$$24x^{2}+5x-360=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 24 for a, 5 for b, and -360 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-5±\sqrt{5^{2}-4\times 24\left(-360\right)}}{2\times 24}$$

Square 5.

$$x=\frac{-5±\sqrt{25-4\times 24\left(-360\right)}}{2\times 24}$$

Multiply -4 times 24.

$$x=\frac{-5±\sqrt{25-96\left(-360\right)}}{2\times 24}$$

Multiply -96 times -360.

$$x=\frac{-5±\sqrt{25+34560}}{2\times 24}$$

Add 25 to 34560.

$$x=\frac{-5±\sqrt{34585}}{2\times 24}$$

Multiply 2 times 24.

$$x=\frac{-5±\sqrt{34585}}{48}$$

Now solve the equation x=\frac{-5±\sqrt{34585}}{48} when ± is plus. Add -5 to \sqrt{34585}.

$$x=\frac{\sqrt{34585}-5}{48}$$

Now solve the equation x=\frac{-5±\sqrt{34585}}{48} when ± is minus. Subtract \sqrt{34585} from -5.

$$x=\frac{-\sqrt{34585}-5}{48}$$

The equation is now solved.

$$x=\frac{\sqrt{34585}-5}{48} x=\frac{-\sqrt{34585}-5}{48}$$

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.

$$24x^{2}+5x=360$$

Divide both sides by 24.

$$\frac{24x^{2}+5x}{24}=\frac{360}{24}$$

Dividing by 24 undoes the multiplication by 24.

$$x^{2}+\frac{5}{24}x=\frac{360}{24}$$

Divide 360 by 24.

$$x^{2}+\frac{5}{24}x=15$$

Divide \frac{5}{24}, the coefficient of the x term, by 2 to get \frac{5}{48}. Then add the square of \frac{5}{48} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{5}{24}x+\left(\frac{5}{48}\right)^{2}=15+\left(\frac{5}{48}\right)^{2}$$

Square \frac{5}{48} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{5}{24}x+\frac{25}{2304}=15+\frac{25}{2304}$$

Add 15 to \frac{25}{2304}.

$$x^{2}+\frac{5}{24}x+\frac{25}{2304}=\frac{34585}{2304}$$

Factor x^{2}+\frac{5}{24}x+\frac{25}{2304}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{5}{48}\right)^{2}=\frac{34585}{2304}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{5}{48}\right)^{2}}=\sqrt{\frac{34585}{2304}}$$

Simplify.

$$x+\frac{5}{48}=\frac{\sqrt{34585}}{48} x+\frac{5}{48}=-\frac{\sqrt{34585}}{48}$$

Subtract \frac{5}{48} from both sides of the equation.

$$x=\frac{\sqrt{34585}-5}{48} x=\frac{-\sqrt{34585}-5}{48}$$