Solve 88x^2-16x=-36

$$88 { x }^{ 2 } -16x=-36$$

Solve for x (complex solution)

$$x=\frac{\sqrt{194}i}{22}+\frac{1}{11}\approx 0.090909091+0.633108558i$$

$$x=-\frac{\sqrt{194}i}{22}+\frac{1}{11}\approx 0.090909091-0.633108558i$$

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.

$$88x^{2}-16x=-36$$

Add 36 to both sides of the equation.

$$88x^{2}-16x-\left(-36\right)=-36-\left(-36\right)$$

Subtracting -36 from itself leaves 0.

$$88x^{2}-16x-\left(-36\right)=0$$

Subtract -36 from 0.

$$88x^{2}-16x+36=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 88 for a, -16 for b, and 36 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-16\right)±\sqrt{\left(-16\right)^{2}-4\times 88\times 36}}{2\times 88}$$

Square -16.

$$x=\frac{-\left(-16\right)±\sqrt{256-4\times 88\times 36}}{2\times 88}$$

Multiply -4 times 88.

$$x=\frac{-\left(-16\right)±\sqrt{256-352\times 36}}{2\times 88}$$

Multiply -352 times 36.

$$x=\frac{-\left(-16\right)±\sqrt{256-12672}}{2\times 88}$$

Add 256 to -12672.

$$x=\frac{-\left(-16\right)±\sqrt{-12416}}{2\times 88}$$

Take the square root of -12416.

$$x=\frac{-\left(-16\right)±8\sqrt{194}i}{2\times 88}$$

The opposite of -16 is 16.

$$x=\frac{16±8\sqrt{194}i}{2\times 88}$$

Multiply 2 times 88.

$$x=\frac{16±8\sqrt{194}i}{176}$$

Now solve the equation x=\frac{16±8\sqrt{194}i}{176} when ± is plus. Add 16 to 8i\sqrt{194}.

$$x=\frac{16+8\sqrt{194}i}{176}$$

Divide 16+8i\sqrt{194} by 176.

$$x=\frac{\sqrt{194}i}{22}+\frac{1}{11}$$

Now solve the equation x=\frac{16±8\sqrt{194}i}{176} when ± is minus. Subtract 8i\sqrt{194} from 16.

$$x=\frac{-8\sqrt{194}i+16}{176}$$

Divide 16-8i\sqrt{194} by 176.

$$x=-\frac{\sqrt{194}i}{22}+\frac{1}{11}$$

The equation is now solved.

$$x=\frac{\sqrt{194}i}{22}+\frac{1}{11} x=-\frac{\sqrt{194}i}{22}+\frac{1}{11}$$

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.

$$88x^{2}-16x=-36$$

Divide both sides by 88.

$$\frac{88x^{2}-16x}{88}=-\frac{36}{88}$$

Dividing by 88 undoes the multiplication by 88.

$$x^{2}+\left(-\frac{16}{88}\right)x=-\frac{36}{88}$$

Reduce the fraction \frac{-16}{88} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{2}{11}x=-\frac{36}{88}$$

Reduce the fraction \frac{-36}{88} to lowest terms by extracting and canceling out 4.

$$x^{2}-\frac{2}{11}x=-\frac{9}{22}$$

Divide -\frac{2}{11}, the coefficient of the x term, by 2 to get -\frac{1}{11}. Then add the square of -\frac{1}{11} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{2}{11}x+\left(-\frac{1}{11}\right)^{2}=-\frac{9}{22}+\left(-\frac{1}{11}\right)^{2}$$

Square -\frac{1}{11} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{2}{11}x+\frac{1}{121}=-\frac{9}{22}+\frac{1}{121}$$

Add -\frac{9}{22} to \frac{1}{121} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{2}{11}x+\frac{1}{121}=-\frac{97}{242}$$

Factor x^{2}-\frac{2}{11}x+\frac{1}{121}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{1}{11}\right)^{2}=-\frac{97}{242}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{11}\right)^{2}}=\sqrt{-\frac{97}{242}}$$

Simplify.

$$x-\frac{1}{11}=\frac{\sqrt{194}i}{22} x-\frac{1}{11}=-\frac{\sqrt{194}i}{22}$$

Add \frac{1}{11} to both sides of the equation.

$$x=\frac{\sqrt{194}i}{22}+\frac{1}{11} x=-\frac{\sqrt{194}i}{22}+\frac{1}{11}$$