Solve 5x^2+74x+240=138

$$5 { x }^{ 2 } +74x+240 = 138$$

Solve for x

$$x=\frac{\sqrt{859}-37}{5}\approx -1.538259644$$

$$x=\frac{-\sqrt{859}-37}{5}\approx -13.261740356$$

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.

$$5x^{2}+74x+240=138$$

Subtract 138 from both sides of the equation.

$$5x^{2}+74x+240-138=138-138$$

Subtracting 138 from itself leaves 0.

$$5x^{2}+74x+240-138=0$$

Subtract 138 from 240.

$$5x^{2}+74x+102=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 5 for a, 74 for b, and 102 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-74±\sqrt{74^{2}-4\times 5\times 102}}{2\times 5}$$

Square 74.

$$x=\frac{-74±\sqrt{5476-4\times 5\times 102}}{2\times 5}$$

Multiply -4 times 5.

$$x=\frac{-74±\sqrt{5476-20\times 102}}{2\times 5}$$

Multiply -20 times 102.

$$x=\frac{-74±\sqrt{5476-2040}}{2\times 5}$$

Add 5476 to -2040.

$$x=\frac{-74±\sqrt{3436}}{2\times 5}$$

Take the square root of 3436.

$$x=\frac{-74±2\sqrt{859}}{2\times 5}$$

Multiply 2 times 5.

$$x=\frac{-74±2\sqrt{859}}{10}$$

Now solve the equation x=\frac{-74±2\sqrt{859}}{10} when ± is plus. Add -74 to 2\sqrt{859}.

$$x=\frac{2\sqrt{859}-74}{10}$$

Divide -74+2\sqrt{859} by 10.

$$x=\frac{\sqrt{859}-37}{5}$$

Now solve the equation x=\frac{-74±2\sqrt{859}}{10} when ± is minus. Subtract 2\sqrt{859} from -74.

$$x=\frac{-2\sqrt{859}-74}{10}$$

Divide -74-2\sqrt{859} by 10.

$$x=\frac{-\sqrt{859}-37}{5}$$

The equation is now solved.

$$x=\frac{\sqrt{859}-37}{5} x=\frac{-\sqrt{859}-37}{5}$$

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.

$$5x^{2}+74x+240=138$$

Subtract 240 from both sides of the equation.

$$5x^{2}+74x+240-240=138-240$$

Subtracting 240 from itself leaves 0.

$$5x^{2}+74x=138-240$$

Subtract 240 from 138.

$$5x^{2}+74x=-102$$

Divide both sides by 5.

$$\frac{5x^{2}+74x}{5}=-\frac{102}{5}$$

Dividing by 5 undoes the multiplication by 5.

$$x^{2}+\frac{74}{5}x=-\frac{102}{5}$$

Divide \frac{74}{5}, the coefficient of the x term, by 2 to get \frac{37}{5}. Then add the square of \frac{37}{5} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{74}{5}x+\left(\frac{37}{5}\right)^{2}=-\frac{102}{5}+\left(\frac{37}{5}\right)^{2}$$

Square \frac{37}{5} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{74}{5}x+\frac{1369}{25}=-\frac{102}{5}+\frac{1369}{25}$$

Add -\frac{102}{5} to \frac{1369}{25} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{74}{5}x+\frac{1369}{25}=\frac{859}{25}$$

Factor x^{2}+\frac{74}{5}x+\frac{1369}{25}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{37}{5}\right)^{2}=\frac{859}{25}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{37}{5}\right)^{2}}=\sqrt{\frac{859}{25}}$$

Simplify.

$$x+\frac{37}{5}=\frac{\sqrt{859}}{5} x+\frac{37}{5}=-\frac{\sqrt{859}}{5}$$

Subtract \frac{37}{5} from both sides of the equation.

$$x=\frac{\sqrt{859}-37}{5} x=\frac{-\sqrt{859}-37}{5}$$