Solve 4x^2=36x-87

$$4 x ^ { 2 } = 36 x - 87$$

Solve for x (complex solution)

$$x=\frac{9+\sqrt{6}i}{2}\approx 4.5+1.224744871i$$

$$x=\frac{-\sqrt{6}i+9}{2}\approx 4.5-1.224744871i$$

Steps Using the Quadratic Formula

Subtract 36x from both sides.

$$4x^{2}-36x=-87$$

Add 87 to both sides.

$$4x^{2}-36x+87=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 4 for a, -36 for b, and 87 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-36\right)±\sqrt{\left(-36\right)^{2}-4\times 4\times 87}}{2\times 4}$$

Square -36.

$$x=\frac{-\left(-36\right)±\sqrt{1296-4\times 4\times 87}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-\left(-36\right)±\sqrt{1296-16\times 87}}{2\times 4}$$

Multiply -16 times 87.

$$x=\frac{-\left(-36\right)±\sqrt{1296-1392}}{2\times 4}$$

Add 1296 to -1392.

$$x=\frac{-\left(-36\right)±\sqrt{-96}}{2\times 4}$$

Take the square root of -96.

$$x=\frac{-\left(-36\right)±4\sqrt{6}i}{2\times 4}$$

The opposite of -36 is 36.

$$x=\frac{36±4\sqrt{6}i}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{36±4\sqrt{6}i}{8}$$

Now solve the equation x=\frac{36±4\sqrt{6}i}{8} when ± is plus. Add 36 to 4i\sqrt{6}.

$$x=\frac{36+4\sqrt{6}i}{8}$$

Divide 36+4i\sqrt{6} by 8.

$$x=\frac{9+\sqrt{6}i}{2}$$

Now solve the equation x=\frac{36±4\sqrt{6}i}{8} when ± is minus. Subtract 4i\sqrt{6} from 36.

$$x=\frac{-4\sqrt{6}i+36}{8}$$

Divide 36-4i\sqrt{6} by 8.

$$x=\frac{-\sqrt{6}i+9}{2}$$

The equation is now solved.

$$x=\frac{9+\sqrt{6}i}{2} x=\frac{-\sqrt{6}i+9}{2}$$

Steps for Completing the Square

Subtract 36x from both sides.

$$4x^{2}-36x=-87$$

Divide both sides by 4.

$$\frac{4x^{2}-36x}{4}=-\frac{87}{4}$$

Dividing by 4 undoes the multiplication by 4.

$$x^{2}+\left(-\frac{36}{4}\right)x=-\frac{87}{4}$$

Divide -36 by 4.

$$x^{2}-9x=-\frac{87}{4}$$

Divide -9, the coefficient of the x term, by 2 to get -\frac{9}{2}. Then add the square of -\frac{9}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-9x+\left(-\frac{9}{2}\right)^{2}=-\frac{87}{4}+\left(-\frac{9}{2}\right)^{2}$$

Square -\frac{9}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-9x+\frac{81}{4}=\frac{-87+81}{4}$$

Add -\frac{87}{4} to \frac{81}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-9x+\frac{81}{4}=-\frac{3}{2}$$

Factor x^{2}-9x+\frac{81}{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}.

$$\left(x-\frac{9}{2}\right)^{2}=-\frac{3}{2}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{9}{2}\right)^{2}}=\sqrt{-\frac{3}{2}}$$

Simplify.

$$x-\frac{9}{2}=\frac{\sqrt{6}i}{2} x-\frac{9}{2}=-\frac{\sqrt{6}i}{2}$$

Add \frac{9}{2} to both sides of the equation.

$$x=\frac{9+\sqrt{6}i}{2} x=\frac{-\sqrt{6}i+9}{2}$$