Solve 90x^2-90x+16=0

$$90 { x }^{ 2 } -90x+16=0$$

Solve for x

$$x=\frac{\sqrt{65}}{30}+\frac{1}{2}\approx 0.768741925$$

$$x=-\frac{\sqrt{65}}{30}+\frac{1}{2}\approx 0.231258075$$

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.

$$90x^{2}-90x+16=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 90 for a, -90 for b, and 16 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-90\right)±\sqrt{\left(-90\right)^{2}-4\times 90\times 16}}{2\times 90}$$

Square -90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-4\times 90\times 16}}{2\times 90}$$

Multiply -4 times 90.

$$x=\frac{-\left(-90\right)±\sqrt{8100-360\times 16}}{2\times 90}$$

Multiply -360 times 16.

$$x=\frac{-\left(-90\right)±\sqrt{8100-5760}}{2\times 90}$$

Add 8100 to -5760.

$$x=\frac{-\left(-90\right)±\sqrt{2340}}{2\times 90}$$

Take the square root of 2340.

$$x=\frac{-\left(-90\right)±6\sqrt{65}}{2\times 90}$$

The opposite of -90 is 90.

$$x=\frac{90±6\sqrt{65}}{2\times 90}$$

Multiply 2 times 90.

$$x=\frac{90±6\sqrt{65}}{180}$$

Now solve the equation x=\frac{90±6\sqrt{65}}{180} when ± is plus. Add 90 to 6\sqrt{65}.

$$x=\frac{6\sqrt{65}+90}{180}$$

Divide 90+6\sqrt{65} by 180.

$$x=\frac{\sqrt{65}}{30}+\frac{1}{2}$$

Now solve the equation x=\frac{90±6\sqrt{65}}{180} when ± is minus. Subtract 6\sqrt{65} from 90.

$$x=\frac{90-6\sqrt{65}}{180}$$

Divide 90-6\sqrt{65} by 180.

$$x=-\frac{\sqrt{65}}{30}+\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{65}}{30}+\frac{1}{2} x=-\frac{\sqrt{65}}{30}+\frac{1}{2}$$

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.

$$90x^{2}-90x+16=0$$

Subtract 16 from both sides of the equation.

$$90x^{2}-90x+16-16=-16$$

Subtracting 16 from itself leaves 0.

$$90x^{2}-90x=-16$$

Divide both sides by 90.

$$\frac{90x^{2}-90x}{90}=-\frac{16}{90}$$

Dividing by 90 undoes the multiplication by 90.

$$x^{2}+\left(-\frac{90}{90}\right)x=-\frac{16}{90}$$

Divide -90 by 90.

$$x^{2}-x=-\frac{16}{90}$$

Reduce the fraction \frac{-16}{90} to lowest terms by extracting and canceling out 2.

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

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

$$x^{2}-x+\left(-\frac{1}{2}\right)^{2}=-\frac{8}{45}+\left(-\frac{1}{2}\right)^{2}$$

Square -\frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-x+\frac{1}{4}=-\frac{8}{45}+\frac{1}{4}$$

Add -\frac{8}{45} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-x+\frac{1}{4}=\frac{13}{180}$$

Factor x^{2}-x+\frac{1}{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}.

$$\left(x-\frac{1}{2}\right)^{2}=\frac{13}{180}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{1}{2}\right)^{2}}=\sqrt{\frac{13}{180}}$$

Simplify.

$$x-\frac{1}{2}=\frac{\sqrt{65}}{30} x-\frac{1}{2}=-\frac{\sqrt{65}}{30}$$

Add \frac{1}{2} to both sides of the equation.

$$x=\frac{\sqrt{65}}{30}+\frac{1}{2} x=-\frac{\sqrt{65}}{30}+\frac{1}{2}$$