Solve 8a^2-16a=6a-12

$$8 a ^ { 2 } - 16 a = 6 a - 12$$

Solve for a

$$a=2$$

$$a=\frac{3}{4}=0.75$$

Steps Using Factoring By Grouping

Subtract 6a from both sides.

$$8a^{2}-16a-6a=-12$$

Combine -16a and -6a to get -22a.

$$8a^{2}-22a=-12$$

Add 12 to both sides.

$$8a^{2}-22a+12=0$$

Divide both sides by 2.

$$4a^{2}-11a+6=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 4a^{2}+aa+ba+6. To find a and b, set up a system to be solved.

$$a+b=-11 ab=4\times 6=24$$

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 24.

$$-1,-24 -2,-12 -3,-8 -4,-6$$

Calculate the sum for each pair.

$$-1-24=-25 -2-12=-14 -3-8=-11 -4-6=-10$$

The solution is the pair that gives sum -11.

$$a=-8 b=-3$$

Rewrite 4a^{2}-11a+6 as \left(4a^{2}-8a\right)+\left(-3a+6\right).

$$\left(4a^{2}-8a\right)+\left(-3a+6\right)$$

Factor out 4a in the first and -3 in the second group.

$$4a\left(a-2\right)-3\left(a-2\right)$$

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

$$\left(a-2\right)\left(4a-3\right)$$

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

$$a=2 a=\frac{3}{4}$$

Steps Using the Quadratic Formula

Subtract 6a from both sides.

$$8a^{2}-16a-6a=-12$$

Combine -16a and -6a to get -22a.

$$8a^{2}-22a=-12$$

Add 12 to both sides.

$$8a^{2}-22a+12=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 8 for a, -22 for b, and 12 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-22\right)±\sqrt{\left(-22\right)^{2}-4\times 8\times 12}}{2\times 8}$$

Square -22.

$$a=\frac{-\left(-22\right)±\sqrt{484-4\times 8\times 12}}{2\times 8}$$

Multiply -4 times 8.

$$a=\frac{-\left(-22\right)±\sqrt{484-32\times 12}}{2\times 8}$$

Multiply -32 times 12.

$$a=\frac{-\left(-22\right)±\sqrt{484-384}}{2\times 8}$$

Add 484 to -384.

$$a=\frac{-\left(-22\right)±\sqrt{100}}{2\times 8}$$

Take the square root of 100.

$$a=\frac{-\left(-22\right)±10}{2\times 8}$$

The opposite of -22 is 22.

$$a=\frac{22±10}{2\times 8}$$

Multiply 2 times 8.

$$a=\frac{22±10}{16}$$

Now solve the equation a=\frac{22±10}{16} when ± is plus. Add 22 to 10.

$$a=\frac{32}{16}$$

Divide 32 by 16.

$$a=2$$

Now solve the equation a=\frac{22±10}{16} when ± is minus. Subtract 10 from 22.

$$a=\frac{12}{16}$$

Reduce the fraction \frac{12}{16} to lowest terms by extracting and canceling out 4.

$$a=\frac{3}{4}$$

The equation is now solved.

$$a=2 a=\frac{3}{4}$$

Steps for Completing the Square

Subtract 6a from both sides.

$$8a^{2}-16a-6a=-12$$

Combine -16a and -6a to get -22a.

$$8a^{2}-22a=-12$$

Divide both sides by 8.

$$\frac{8a^{2}-22a}{8}=-\frac{12}{8}$$

Dividing by 8 undoes the multiplication by 8.

$$a^{2}+\left(-\frac{22}{8}\right)a=-\frac{12}{8}$$

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

$$a^{2}-\frac{11}{4}a=-\frac{12}{8}$$

Reduce the fraction \frac{-12}{8} to lowest terms by extracting and canceling out 4.

$$a^{2}-\frac{11}{4}a=-\frac{3}{2}$$

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.

$$a^{2}-\frac{11}{4}a+\left(-\frac{11}{8}\right)^{2}=-\frac{3}{2}+\left(-\frac{11}{8}\right)^{2}$$

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

$$a^{2}-\frac{11}{4}a+\frac{121}{64}=-\frac{3}{2}+\frac{121}{64}$$

Add -\frac{3}{2} to \frac{121}{64} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$a^{2}-\frac{11}{4}a+\frac{121}{64}=\frac{25}{64}$$

Factor a^{2}-\frac{11}{4}a+\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}.

$$\left(a-\frac{11}{8}\right)^{2}=\frac{25}{64}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-\frac{11}{8}\right)^{2}}=\sqrt{\frac{25}{64}}$$

Simplify.

$$a-\frac{11}{8}=\frac{5}{8} a-\frac{11}{8}=-\frac{5}{8}$$

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

$$a=2 a=\frac{3}{4}$$