Solve 9x^2+90x=20x

$$9 x ^ { 2 } + 90 x = 20 x$$

Solve for x

$$x = -\frac{70}{9} = -7\frac{7}{9} \approx -7.777777778$$

$$x=0$$

Steps Using Factoring

Subtract 20x from both sides.

$$9x^{2}+90x-20x=0$$

Combine 90x and -20x to get 70x.

$$9x^{2}+70x=0$$

Factor out x.

$$x\left(9x+70\right)=0$$

To find equation solutions, solve x=0 and 9x+70=0.

$$x=0 x=-\frac{70}{9}$$

Steps Using the Quadratic Formula

Subtract 20x from both sides.

$$9x^{2}+90x-20x=0$$

Combine 90x and -20x to get 70x.

$$9x^{2}+70x=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, 70 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-70±\sqrt{70^{2}}}{2\times 9}$$

Take the square root of 70^{2}.

$$x=\frac{-70±70}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-70±70}{18}$$

Now solve the equation x=\frac{-70±70}{18} when ± is plus. Add -70 to 70.

$$x=\frac{0}{18}$$

Divide 0 by 18.

$$x=0$$

Now solve the equation x=\frac{-70±70}{18} when ± is minus. Subtract 70 from -70.

$$x=-\frac{140}{18}$$

Reduce the fraction \frac{-140}{18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{70}{9}$$

The equation is now solved.

$$x=0 x=-\frac{70}{9}$$

Steps for Completing the Square

Subtract 20x from both sides.

$$9x^{2}+90x-20x=0$$

Combine 90x and -20x to get 70x.

$$9x^{2}+70x=0$$

Divide both sides by 9.

$$\frac{9x^{2}+70x}{9}=\frac{0}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}+\frac{70}{9}x=\frac{0}{9}$$

Divide 0 by 9.

$$x^{2}+\frac{70}{9}x=0$$

Divide \frac{70}{9}, the coefficient of the x term, by 2 to get \frac{35}{9}. Then add the square of \frac{35}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}+\frac{70}{9}x+\left(\frac{35}{9}\right)^{2}=\left(\frac{35}{9}\right)^{2}$$

Square \frac{35}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}+\frac{70}{9}x+\frac{1225}{81}=\frac{1225}{81}$$

Factor x^{2}+\frac{70}{9}x+\frac{1225}{81}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+\frac{35}{9}\right)^{2}=\frac{1225}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+\frac{35}{9}\right)^{2}}=\sqrt{\frac{1225}{81}}$$

Simplify.

$$x+\frac{35}{9}=\frac{35}{9} x+\frac{35}{9}=-\frac{35}{9}$$

Subtract \frac{35}{9} from both sides of the equation.

$$x=0 x=-\frac{70}{9}$$