Solve 9x^2=22x-8

$$9 x ^ { 2 } = 22 x - 8$$

Solve for x

$$x=2$$

$$x=\frac{4}{9}\approx 0.444444444$$

Steps Using Factoring By Grouping

Subtract 22x from both sides.

$$9x^{2}-22x=-8$$

Add 8 to both sides.

$$9x^{2}-22x+8=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 9x^{2}+ax+bx+8. To find a and b, set up a system to be solved.

$$a+b=-22 ab=9\times 8=72$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 72.

$$-1,-72 -2,-36 -3,-24 -4,-18 -6,-12 -8,-9$$

Calculate the sum for each pair.

$$-1-72=-73 -2-36=-38 -3-24=-27 -4-18=-22 -6-12=-18 -8-9=-17$$

The solution is the pair that gives sum -22.

$$a=-18 b=-4$$

Rewrite 9x^{2}-22x+8 as \left(9x^{2}-18x\right)+\left(-4x+8\right).

$$\left(9x^{2}-18x\right)+\left(-4x+8\right)$$

Factor out 9x in the first and -4 in the second group.

$$9x\left(x-2\right)-4\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(9x-4\right)$$

To find equation solutions, solve x-2=0 and 9x-4=0.

$$x=2 x=\frac{4}{9}$$

Steps Using the Quadratic Formula

Subtract 22x from both sides.

$$9x^{2}-22x=-8$$

Add 8 to both sides.

$$9x^{2}-22x+8=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 9 for a, -22 for b, and 8 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 9\times 8}}{2\times 9}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 9\times 8}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-22\right)±\sqrt{484-36\times 8}}{2\times 9}$$

Multiply -36 times 8.

$$x=\frac{-\left(-22\right)±\sqrt{484-288}}{2\times 9}$$

Add 484 to -288.

$$x=\frac{-\left(-22\right)±\sqrt{196}}{2\times 9}$$

Take the square root of 196.

$$x=\frac{-\left(-22\right)±14}{2\times 9}$$

The opposite of -22 is 22.

$$x=\frac{22±14}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{22±14}{18}$$

Now solve the equation x=\frac{22±14}{18} when ± is plus. Add 22 to 14.

$$x=\frac{36}{18}$$

Divide 36 by 18.

$$x=2$$

Now solve the equation x=\frac{22±14}{18} when ± is minus. Subtract 14 from 22.

$$x=\frac{8}{18}$$

Reduce the fraction \frac{8}{18} to lowest terms by extracting and canceling out 2.

$$x=\frac{4}{9}$$

The equation is now solved.

$$x=2 x=\frac{4}{9}$$

Steps for Completing the Square

Subtract 22x from both sides.

$$9x^{2}-22x=-8$$

Divide both sides by 9.

$$\frac{9x^{2}-22x}{9}=-\frac{8}{9}$$

Dividing by 9 undoes the multiplication by 9.

$$x^{2}-\frac{22}{9}x=-\frac{8}{9}$$

Divide -\frac{22}{9}, the coefficient of the x term, by 2 to get -\frac{11}{9}. Then add the square of -\frac{11}{9} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{22}{9}x+\left(-\frac{11}{9}\right)^{2}=-\frac{8}{9}+\left(-\frac{11}{9}\right)^{2}$$

Square -\frac{11}{9} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{22}{9}x+\frac{121}{81}=-\frac{8}{9}+\frac{121}{81}$$

Add -\frac{8}{9} to \frac{121}{81} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{22}{9}x+\frac{121}{81}=\frac{49}{81}$$

Factor x^{2}-\frac{22}{9}x+\frac{121}{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}.

$$\left(x-\frac{11}{9}\right)^{2}=\frac{49}{81}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{9}\right)^{2}}=\sqrt{\frac{49}{81}}$$

Simplify.

$$x-\frac{11}{9}=\frac{7}{9} x-\frac{11}{9}=-\frac{7}{9}$$

Add \frac{11}{9} to both sides of the equation.

$$x=2 x=\frac{4}{9}$$