Solve 9x^2-35x=150

$$9 x ^ { 2 } - 35 x = 150$$

Solve for x

$$x = \frac{5 \sqrt{265} + 35}{18} \approx 6.466339054$$

$$x=\frac{35-5\sqrt{265}}{18}\approx -2.577450166$$

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.

$$9x^{2}-35x=150$$

Subtract 150 from both sides of the equation.

$$9x^{2}-35x-150=150-150$$

Subtracting 150 from itself leaves 0.

$$9x^{2}-35x-150=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -35 for b, and -150 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-35\right)±\sqrt{\left(-35\right)^{2}-4\times 9\left(-150\right)}}{2\times 9}$$

Square -35.

$$x=\frac{-\left(-35\right)±\sqrt{1225-4\times 9\left(-150\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-35\right)±\sqrt{1225-36\left(-150\right)}}{2\times 9}$$

Multiply -36 times -150.

$$x=\frac{-\left(-35\right)±\sqrt{1225+5400}}{2\times 9}$$

Add 1225 to 5400.

$$x=\frac{-\left(-35\right)±\sqrt{6625}}{2\times 9}$$

Take the square root of 6625.

$$x=\frac{-\left(-35\right)±5\sqrt{265}}{2\times 9}$$

The opposite of -35 is 35.

$$x=\frac{35±5\sqrt{265}}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{35±5\sqrt{265}}{18}$$

Now solve the equation x=\frac{35±5\sqrt{265}}{18} when ± is plus. Add 35 to 5\sqrt{265}.

$$x=\frac{5\sqrt{265}+35}{18}$$

Now solve the equation x=\frac{35±5\sqrt{265}}{18} when ± is minus. Subtract 5\sqrt{265} from 35.

$$x=\frac{35-5\sqrt{265}}{18}$$

The equation is now solved.

$$x=\frac{5\sqrt{265}+35}{18} x=\frac{35-5\sqrt{265}}{18}$$

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.

$$9x^{2}-35x=150$$

Divide both sides by 9.

$$\frac{9x^{2}-35x}{9}=\frac{150}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}-\frac{35}{9}x=\frac{150}{9}$$

Reduce the fraction \frac{150}{9} to lowest terms by extracting and canceling out 3.

$$x^{2}-\frac{35}{9}x=\frac{50}{3}$$

Divide -\frac{35}{9}, the coefficient of the x term, by 2 to get -\frac{35}{18}. Then add the square of -\frac{35}{18} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{35}{9}x+\left(-\frac{35}{18}\right)^{2}=\frac{50}{3}+\left(-\frac{35}{18}\right)^{2}$$

Square -\frac{35}{18} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{35}{9}x+\frac{1225}{324}=\frac{50}{3}+\frac{1225}{324}$$

Add \frac{50}{3} to \frac{1225}{324} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{35}{9}x+\frac{1225}{324}=\frac{6625}{324}$$

Factor x^{2}-\frac{35}{9}x+\frac{1225}{324}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{35}{18}\right)^{2}=\frac{6625}{324}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{35}{18}\right)^{2}}=\sqrt{\frac{6625}{324}}$$

Simplify.

$$x-\frac{35}{18}=\frac{5\sqrt{265}}{18} x-\frac{35}{18}=-\frac{5\sqrt{265}}{18}$$

Add \frac{35}{18} to both sides of the equation.

$$x=\frac{5\sqrt{265}+35}{18} x=\frac{35-5\sqrt{265}}{18}$$