Solve 45x^2+27x+32=15

$$45 { x }^{ 2 } +27x+32=15$$

Solve for x (complex solution)

$$x=\frac{\sqrt{259}i}{30}-\frac{3}{10}\approx -0.3+0.536449231i$$

$$x=-\frac{\sqrt{259}i}{30}-\frac{3}{10}\approx -0.3-0.536449231i$$

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.

$$45x^{2}+27x+32=15$$

Subtract 15 from both sides of the equation.

$$45x^{2}+27x+32-15=15-15$$

Subtracting 15 from itself leaves 0.

$$45x^{2}+27x+32-15=0$$

Subtract 15 from 32.

$$45x^{2}+27x+17=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 45 for a, 27 for b, and 17 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-27±\sqrt{27^{2}-4\times 45\times 17}}{2\times 45}$$

Square 27.

$$x=\frac{-27±\sqrt{729-4\times 45\times 17}}{2\times 45}$$

Multiply -4 times 45.

$$x=\frac{-27±\sqrt{729-180\times 17}}{2\times 45}$$

Multiply -180 times 17.

$$x=\frac{-27±\sqrt{729-3060}}{2\times 45}$$

Add 729 to -3060.

$$x=\frac{-27±\sqrt{-2331}}{2\times 45}$$

Take the square root of -2331.

$$x=\frac{-27±3\sqrt{259}i}{2\times 45}$$

Multiply 2 times 45.

$$x=\frac{-27±3\sqrt{259}i}{90}$$

Now solve the equation x=\frac{-27±3\sqrt{259}i}{90} when ± is plus. Add -27 to 3i\sqrt{259}.

$$x=\frac{-27+3\sqrt{259}i}{90}$$

Divide -27+3i\sqrt{259} by 90.

$$x=\frac{\sqrt{259}i}{30}-\frac{3}{10}$$

Now solve the equation x=\frac{-27±3\sqrt{259}i}{90} when ± is minus. Subtract 3i\sqrt{259} from -27.

$$x=\frac{-3\sqrt{259}i-27}{90}$$

Divide -27-3i\sqrt{259} by 90.

$$x=-\frac{\sqrt{259}i}{30}-\frac{3}{10}$$

The equation is now solved.

$$x=\frac{\sqrt{259}i}{30}-\frac{3}{10} x=-\frac{\sqrt{259}i}{30}-\frac{3}{10}$$

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.

$$45x^{2}+27x+32=15$$

Subtract 32 from both sides of the equation.

$$45x^{2}+27x+32-32=15-32$$

Subtracting 32 from itself leaves 0.

$$45x^{2}+27x=15-32$$

Subtract 32 from 15.

$$45x^{2}+27x=-17$$

Divide both sides by 45.

$$\frac{45x^{2}+27x}{45}=-\frac{17}{45}$$

Dividing by 45 undoes the multiplication by 45.

$$x^{2}+\frac{27}{45}x=-\frac{17}{45}$$

Reduce the fraction \frac{27}{45} to lowest terms by extracting and canceling out 9.

$$x^{2}+\frac{3}{5}x=-\frac{17}{45}$$

Divide \frac{3}{5}, the coefficient of the x term, by 2 to get \frac{3}{10}. Then add the square of \frac{3}{10} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{3}{5}x+\left(\frac{3}{10}\right)^{2}=-\frac{17}{45}+\left(\frac{3}{10}\right)^{2}$$

Square \frac{3}{10} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=-\frac{17}{45}+\frac{9}{100}$$

Add -\frac{17}{45} to \frac{9}{100} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+\frac{3}{5}x+\frac{9}{100}=-\frac{259}{900}$$

Factor x^{2}+\frac{3}{5}x+\frac{9}{100}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{3}{10}\right)^{2}=-\frac{259}{900}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{3}{10}\right)^{2}}=\sqrt{-\frac{259}{900}}$$

Simplify.

$$x+\frac{3}{10}=\frac{\sqrt{259}i}{30} x+\frac{3}{10}=-\frac{\sqrt{259}i}{30}$$

Subtract \frac{3}{10} from both sides of the equation.

$$x=\frac{\sqrt{259}i}{30}-\frac{3}{10} x=-\frac{\sqrt{259}i}{30}-\frac{3}{10}$$