Solve (9)5x^2-8x-13=0

$$( 9 ) 5 x ^ { 2 } - 8 x - 13 = 0$$

Solve for x

$$x=\frac{\sqrt{601}+4}{45}\approx 0.633673363$$

$$x=\frac{4-\sqrt{601}}{45}\approx -0.455895585$$

Steps Using the Quadratic Formula

Multiply 9 and 5 to get 45.

$$45x^{2}-8x-13=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 45 for a, -8 for b, and -13 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\times 45\left(-13\right)}}{2\times 45}$$

Square -8.

$$x=\frac{-\left(-8\right)±\sqrt{64-4\times 45\left(-13\right)}}{2\times 45}$$

Multiply -4 times 45.

$$x=\frac{-\left(-8\right)±\sqrt{64-180\left(-13\right)}}{2\times 45}$$

Multiply -180 times -13.

$$x=\frac{-\left(-8\right)±\sqrt{64+2340}}{2\times 45}$$

Add 64 to 2340.

$$x=\frac{-\left(-8\right)±\sqrt{2404}}{2\times 45}$$

Take the square root of 2404.

$$x=\frac{-\left(-8\right)±2\sqrt{601}}{2\times 45}$$

The opposite of -8 is 8.

$$x=\frac{8±2\sqrt{601}}{2\times 45}$$

Multiply 2 times 45.

$$x=\frac{8±2\sqrt{601}}{90}$$

Now solve the equation x=\frac{8±2\sqrt{601}}{90} when ± is plus. Add 8 to 2\sqrt{601}.

$$x=\frac{2\sqrt{601}+8}{90}$$

Divide 8+2\sqrt{601} by 90.

$$x=\frac{\sqrt{601}+4}{45}$$

Now solve the equation x=\frac{8±2\sqrt{601}}{90} when ± is minus. Subtract 2\sqrt{601} from 8.

$$x=\frac{8-2\sqrt{601}}{90}$$

Divide 8-2\sqrt{601} by 90.

$$x=\frac{4-\sqrt{601}}{45}$$

The equation is now solved.

$$x=\frac{\sqrt{601}+4}{45} x=\frac{4-\sqrt{601}}{45}$$

Steps for Completing the Square

Multiply 9 and 5 to get 45.

$$45x^{2}-8x-13=0$$

Add 13 to both sides. Anything plus zero gives itself.

$$45x^{2}-8x=13$$

Divide both sides by 45.

$$\frac{45x^{2}-8x}{45}=\frac{13}{45}$$

Dividing by 45 undoes the multiplication by 45.

$$x^{2}-\frac{8}{45}x=\frac{13}{45}$$

Divide -\frac{8}{45}, the coefficient of the x term, by 2 to get -\frac{4}{45}. Then add the square of -\frac{4}{45} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{8}{45}x+\left(-\frac{4}{45}\right)^{2}=\frac{13}{45}+\left(-\frac{4}{45}\right)^{2}$$

Square -\frac{4}{45} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{8}{45}x+\frac{16}{2025}=\frac{13}{45}+\frac{16}{2025}$$

Add \frac{13}{45} to \frac{16}{2025} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{8}{45}x+\frac{16}{2025}=\frac{601}{2025}$$

Factor x^{2}-\frac{8}{45}x+\frac{16}{2025}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{4}{45}\right)^{2}=\frac{601}{2025}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{4}{45}\right)^{2}}=\sqrt{\frac{601}{2025}}$$

Simplify.

$$x-\frac{4}{45}=\frac{\sqrt{601}}{45} x-\frac{4}{45}=-\frac{\sqrt{601}}{45}$$

Add \frac{4}{45} to both sides of the equation.

$$x=\frac{\sqrt{601}+4}{45} x=\frac{4-\sqrt{601}}{45}$$