Solve 150x^2+150x-66=0

$$150 { x }^{ 2 } +150x-66=0$$

Solve for x

$$x=\frac{\sqrt{69}}{10}-\frac{1}{2}\approx 0.330662386$$

$$x=-\frac{\sqrt{69}}{10}-\frac{1}{2}\approx -1.330662386$$

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.

$$150x^{2}+150x-66=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 150 for a, 150 for b, and -66 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-150±\sqrt{150^{2}-4\times 150\left(-66\right)}}{2\times 150}$$

Square 150.

$$x=\frac{-150±\sqrt{22500-4\times 150\left(-66\right)}}{2\times 150}$$

Multiply -4 times 150.

$$x=\frac{-150±\sqrt{22500-600\left(-66\right)}}{2\times 150}$$

Multiply -600 times -66.

$$x=\frac{-150±\sqrt{22500+39600}}{2\times 150}$$

Add 22500 to 39600.

$$x=\frac{-150±\sqrt{62100}}{2\times 150}$$

Take the square root of 62100.

$$x=\frac{-150±30\sqrt{69}}{2\times 150}$$

Multiply 2 times 150.

$$x=\frac{-150±30\sqrt{69}}{300}$$

Now solve the equation x=\frac{-150±30\sqrt{69}}{300} when ± is plus. Add -150 to 30\sqrt{69}.

$$x=\frac{30\sqrt{69}-150}{300}$$

Divide -150+30\sqrt{69} by 300.

$$x=\frac{\sqrt{69}}{10}-\frac{1}{2}$$

Now solve the equation x=\frac{-150±30\sqrt{69}}{300} when ± is minus. Subtract 30\sqrt{69} from -150.

$$x=\frac{-30\sqrt{69}-150}{300}$$

Divide -150-30\sqrt{69} by 300.

$$x=-\frac{\sqrt{69}}{10}-\frac{1}{2}$$

The equation is now solved.

$$x=\frac{\sqrt{69}}{10}-\frac{1}{2} x=-\frac{\sqrt{69}}{10}-\frac{1}{2}$$

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.

$$150x^{2}+150x-66=0$$

Add 66 to both sides of the equation.

$$150x^{2}+150x-66-\left(-66\right)=-\left(-66\right)$$

Subtracting -66 from itself leaves 0.

$$150x^{2}+150x=-\left(-66\right)$$

Subtract -66 from 0.

$$150x^{2}+150x=66$$

Divide both sides by 150.

$$\frac{150x^{2}+150x}{150}=\frac{66}{150}$$

Dividing by 150 undoes the multiplication by 150.

$$x^{2}+\frac{150}{150}x=\frac{66}{150}$$

Divide 150 by 150.

$$x^{2}+x=\frac{66}{150}$$

Reduce the fraction \frac{66}{150} to lowest terms by extracting and canceling out 6.

$$x^{2}+x=\frac{11}{25}$$

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.

$$x^{2}+x+\left(\frac{1}{2}\right)^{2}=\frac{11}{25}+\left(\frac{1}{2}\right)^{2}$$

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

$$x^{2}+x+\frac{1}{4}=\frac{11}{25}+\frac{1}{4}$$

Add \frac{11}{25} to \frac{1}{4} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}+x+\frac{1}{4}=\frac{69}{100}$$

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}.

$$\left(x+\frac{1}{2}\right)^{2}=\frac{69}{100}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{1}{2}\right)^{2}}=\sqrt{\frac{69}{100}}$$

Simplify.

$$x+\frac{1}{2}=\frac{\sqrt{69}}{10} x+\frac{1}{2}=-\frac{\sqrt{69}}{10}$$

Subtract \frac{1}{2} from both sides of the equation.

$$x=\frac{\sqrt{69}}{10}-\frac{1}{2} x=-\frac{\sqrt{69}}{10}-\frac{1}{2}$$