Solve 8x^2=22x-15

$$8 x ^ { 2 } = 22 x - 15$$

Solve for x

$$x = \frac{3}{2} = 1\frac{1}{2} = 1.5$$

$$x = \frac{5}{4} = 1\frac{1}{4} = 1.25$$

Steps Using Factoring By Grouping

Subtract 22x from both sides.

$$8x^{2}-22x=-15$$

Add 15 to both sides.

$$8x^{2}-22x+15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 8x^{2}+ax+bx+15. To find a and b, set up a system to be solved.

$$a+b=-22 ab=8\times 15=120$$

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 120.

$$-1,-120 -2,-60 -3,-40 -4,-30 -5,-24 -6,-20 -8,-15 -10,-12$$

Calculate the sum for each pair.

$$-1-120=-121 -2-60=-62 -3-40=-43 -4-30=-34 -5-24=-29 -6-20=-26 -8-15=-23 -10-12=-22$$

The solution is the pair that gives sum -22.

$$a=-12 b=-10$$

Rewrite 8x^{2}-22x+15 as \left(8x^{2}-12x\right)+\left(-10x+15\right).

$$\left(8x^{2}-12x\right)+\left(-10x+15\right)$$

Factor out 4x in the first and -5 in the second group.

$$4x\left(2x-3\right)-5\left(2x-3\right)$$

Factor out common term 2x-3 by using distributive property.

$$\left(2x-3\right)\left(4x-5\right)$$

To find equation solutions, solve 2x-3=0 and 4x-5=0.

$$x=\frac{3}{2} x=\frac{5}{4}$$

Steps Using the Quadratic Formula

Subtract 22x from both sides.

$$8x^{2}-22x=-15$$

Add 15 to both sides.

$$8x^{2}-22x+15=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -22 for b, and 15 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 8\times 15}}{2\times 8}$$

Square -22.

$$x=\frac{-\left(-22\right)±\sqrt{484-4\times 8\times 15}}{2\times 8}$$

Multiply -4 times 8.

$$x=\frac{-\left(-22\right)±\sqrt{484-32\times 15}}{2\times 8}$$

Multiply -32 times 15.

$$x=\frac{-\left(-22\right)±\sqrt{484-480}}{2\times 8}$$

Add 484 to -480.

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

Take the square root of 4.

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

The opposite of -22 is 22.

$$x=\frac{22±2}{2\times 8}$$

Multiply 2 times 8.

$$x=\frac{22±2}{16}$$

Now solve the equation x=\frac{22±2}{16} when ± is plus. Add 22 to 2.

$$x=\frac{24}{16}$$

Reduce the fraction \frac{24}{16} to lowest terms by extracting and canceling out 8.

$$x=\frac{3}{2}$$

Now solve the equation x=\frac{22±2}{16} when ± is minus. Subtract 2 from 22.

$$x=\frac{20}{16}$$

Reduce the fraction \frac{20}{16} to lowest terms by extracting and canceling out 4.

$$x=\frac{5}{4}$$

The equation is now solved.

$$x=\frac{3}{2} x=\frac{5}{4}$$

Steps for Completing the Square

Subtract 22x from both sides.

$$8x^{2}-22x=-15$$

Divide both sides by 8.

$$\frac{8x^{2}-22x}{8}=-\frac{15}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$x^{2}+\left(-\frac{22}{8}\right)x=-\frac{15}{8}$$

Reduce the fraction \frac{-22}{8} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{11}{4}x=-\frac{15}{8}$$

Divide -\frac{11}{4}, the coefficient of the x term, by 2 to get -\frac{11}{8}. Then add the square of -\frac{11}{8} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{11}{4}x+\left(-\frac{11}{8}\right)^{2}=-\frac{15}{8}+\left(-\frac{11}{8}\right)^{2}$$

Square -\frac{11}{8} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=-\frac{15}{8}+\frac{121}{64}$$

Add -\frac{15}{8} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{11}{4}x+\frac{121}{64}=\frac{1}{64}$$

Factor x^{2}-\frac{11}{4}x+\frac{121}{64}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{11}{8}\right)^{2}=\frac{1}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{11}{8}\right)^{2}}=\sqrt{\frac{1}{64}}$$

Simplify.

$$x-\frac{11}{8}=\frac{1}{8} x-\frac{11}{8}=-\frac{1}{8}$$

Add \frac{11}{8} to both sides of the equation.

$$x=\frac{3}{2} x=\frac{5}{4}$$