Solve 40x^2+120x-15=0

$$40 { x }^{ 2 } +120x-15=0$$

Solve for x

$$x=\frac{\sqrt{42}}{4}-\frac{3}{2}\approx 0.120185175$$

$$x=-\frac{\sqrt{42}}{4}-\frac{3}{2}\approx -3.120185175$$

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.

$$40x^{2}+120x-15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 40 for a, 120 for b, and -15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-120±\sqrt{120^{2}-4\times 40\left(-15\right)}}{2\times 40}$$

Square 120.

$$x=\frac{-120±\sqrt{14400-4\times 40\left(-15\right)}}{2\times 40}$$

Multiply -4 times 40.

$$x=\frac{-120±\sqrt{14400-160\left(-15\right)}}{2\times 40}$$

Multiply -160 times -15.

$$x=\frac{-120±\sqrt{14400+2400}}{2\times 40}$$

Add 14400 to 2400.

$$x=\frac{-120±\sqrt{16800}}{2\times 40}$$

Take the square root of 16800.

$$x=\frac{-120±20\sqrt{42}}{2\times 40}$$

Multiply 2 times 40.

$$x=\frac{-120±20\sqrt{42}}{80}$$

Now solve the equation x=\frac{-120±20\sqrt{42}}{80} when ± is plus. Add -120 to 20\sqrt{42}.

$$x=\frac{20\sqrt{42}-120}{80}$$

Divide -120+20\sqrt{42} by 80.

$$x=\frac{\sqrt{42}}{4}-\frac{3}{2}$$

Now solve the equation x=\frac{-120±20\sqrt{42}}{80} when ± is minus. Subtract 20\sqrt{42} from -120.

$$x=\frac{-20\sqrt{42}-120}{80}$$

Divide -120-20\sqrt{42} by 80.

$$x=-\frac{\sqrt{42}}{4}-\frac{3}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{42}}{4}-\frac{3}{2} x=-\frac{\sqrt{42}}{4}-\frac{3}{2}$$

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.

$$40x^{2}+120x-15=0$$

Add 15 to both sides of the equation.

$$40x^{2}+120x-15-\left(-15\right)=-\left(-15\right)$$

Subtracting -15 from itself leaves 0.

$$40x^{2}+120x=-\left(-15\right)$$

Subtract -15 from 0.

$$40x^{2}+120x=15$$

Divide both sides by 40.

$$\frac{40x^{2}+120x}{40}=\frac{15}{40}$$

Dividing by 40 undoes the multiplication by 40.

$$x^{2}+\frac{120}{40}x=\frac{15}{40}$$

Divide 120 by 40.

$$x^{2}+3x=\frac{15}{40}$$

Reduce the fraction \frac{15}{40} to lowest terms by extracting and canceling out 5.

$$x^{2}+3x=\frac{3}{8}$$

Divide 3, the coefficient of the x term, by 2 to get \frac{3}{2}. Then add the square of \frac{3}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+3x+\left(\frac{3}{2}\right)^{2}=\frac{3}{8}+\left(\frac{3}{2}\right)^{2}$$

Square \frac{3}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+3x+\frac{9}{4}=\frac{3}{8}+\frac{9}{4}$$

Add \frac{3}{8} to \frac{9}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+3x+\frac{9}{4}=\frac{21}{8}$$

Factor x^{2}+3x+\frac{9}{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}.

$$\left(x+\frac{3}{2}\right)^{2}=\frac{21}{8}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{2}\right)^{2}}=\sqrt{\frac{21}{8}}$$

Simplify.

$$x+\frac{3}{2}=\frac{\sqrt{42}}{4} x+\frac{3}{2}=-\frac{\sqrt{42}}{4}$$

Subtract \frac{3}{2} from both sides of the equation.

$$x=\frac{\sqrt{42}}{4}-\frac{3}{2} x=-\frac{\sqrt{42}}{4}-\frac{3}{2}$$