Solve (8x+3)(8x-8)-16x-6=0

$$( 8 x + 3 ) ( 8 x - 8 ) - 16 x - 6 = 0$$

Solve for x

$$x=-\frac{3}{8}=-0.375$$

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

Steps Using Factoring By Grouping

Use the distributive property to multiply 8x+3 by 8x-8 and combine like terms.

$$64x^{2}-40x-24-16x-6=0$$

Combine -40x and -16x to get -56x.

$$64x^{2}-56x-24-6=0$$

Subtract 6 from -24 to get -30.

$$64x^{2}-56x-30=0$$

Divide both sides by 2.

$$32x^{2}-28x-15=0$$

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as 32x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=-28 ab=32\left(-15\right)=-480$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -480.

$$1,-480 2,-240 3,-160 4,-120 5,-96 6,-80 8,-60 10,-48 12,-40 15,-32 16,-30 20,-24$$

Calculate the sum for each pair.

$$1-480=-479 2-240=-238 3-160=-157 4-120=-116 5-96=-91 6-80=-74 8-60=-52 10-48=-38 12-40=-28 15-32=-17 16-30=-14 20-24=-4$$

The solution is the pair that gives sum -28.

$$a=-40 b=12$$

Rewrite 32x^{2}-28x-15 as \left(32x^{2}-40x\right)+\left(12x-15\right).

$$\left(32x^{2}-40x\right)+\left(12x-15\right)$$

Factor out 8x in the first and 3 in the second group.

$$8x\left(4x-5\right)+3\left(4x-5\right)$$

Factor out common term 4x-5 by using distributive property.

$$\left(4x-5\right)\left(8x+3\right)$$

To find equation solutions, solve 4x-5=0 and 8x+3=0.

$$x=\frac{5}{4} x=-\frac{3}{8}$$

Steps Using the Quadratic Formula

Use the distributive property to multiply 8x+3 by 8x-8 and combine like terms.

$$64x^{2}-40x-24-16x-6=0$$

Combine -40x and -16x to get -56x.

$$64x^{2}-56x-24-6=0$$

Subtract 6 from -24 to get -30.

$$64x^{2}-56x-30=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 64 for a, -56 for b, and -30 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-\left(-56\right)±\sqrt{\left(-56\right)^{2}-4\times 64\left(-30\right)}}{2\times 64}$$

Square -56.

$$x=\frac{-\left(-56\right)±\sqrt{3136-4\times 64\left(-30\right)}}{2\times 64}$$

Multiply -4 times 64.

$$x=\frac{-\left(-56\right)±\sqrt{3136-256\left(-30\right)}}{2\times 64}$$

Multiply -256 times -30.

$$x=\frac{-\left(-56\right)±\sqrt{3136+7680}}{2\times 64}$$

Add 3136 to 7680.

$$x=\frac{-\left(-56\right)±\sqrt{10816}}{2\times 64}$$

Take the square root of 10816.

$$x=\frac{-\left(-56\right)±104}{2\times 64}$$

The opposite of -56 is 56.

$$x=\frac{56±104}{2\times 64}$$

Multiply 2 times 64.

$$x=\frac{56±104}{128}$$

Now solve the equation x=\frac{56±104}{128} when ± is plus. Add 56 to 104.

$$x=\frac{160}{128}$$

Reduce the fraction \frac{160}{128} to lowest terms by extracting and canceling out 32.

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

Now solve the equation x=\frac{56±104}{128} when ± is minus. Subtract 104 from 56.

$$x=-\frac{48}{128}$$

Reduce the fraction \frac{-48}{128} to lowest terms by extracting and canceling out 16.

$$x=-\frac{3}{8}$$

The equation is now solved.

$$x=\frac{5}{4} x=-\frac{3}{8}$$

Steps for Completing the Square

Use the distributive property to multiply 8x+3 by 8x-8 and combine like terms.

$$64x^{2}-40x-24-16x-6=0$$

Combine -40x and -16x to get -56x.

$$64x^{2}-56x-24-6=0$$

Subtract 6 from -24 to get -30.

$$64x^{2}-56x-30=0$$

Add 30 to both sides. Anything plus zero gives itself.

$$64x^{2}-56x=30$$

Divide both sides by 64.

$$\frac{64x^{2}-56x}{64}=\frac{30}{64}$$

Dividing by 64 undoes the multiplication by 64.

$$x^{2}+\left(-\frac{56}{64}\right)x=\frac{30}{64}$$

Reduce the fraction \frac{-56}{64} to lowest terms by extracting and canceling out 8.

$$x^{2}-\frac{7}{8}x=\frac{30}{64}$$

Reduce the fraction \frac{30}{64} to lowest terms by extracting and canceling out 2.

$$x^{2}-\frac{7}{8}x=\frac{15}{32}$$

Divide -\frac{7}{8}, the coefficient of the x term, by 2 to get -\frac{7}{16}. Then add the square of -\frac{7}{16} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$x^{2}-\frac{7}{8}x+\left(-\frac{7}{16}\right)^{2}=\frac{15}{32}+\left(-\frac{7}{16}\right)^{2}$$

Square -\frac{7}{16} by squaring both the numerator and the denominator of the fraction.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=\frac{15}{32}+\frac{49}{256}$$

Add \frac{15}{32} to \frac{49}{256} by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$x^{2}-\frac{7}{8}x+\frac{49}{256}=\frac{169}{256}$$

Factor x^{2}-\frac{7}{8}x+\frac{49}{256}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x-\frac{7}{16}\right)^{2}=\frac{169}{256}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x-\frac{7}{16}\right)^{2}}=\sqrt{\frac{169}{256}}$$

Simplify.

$$x-\frac{7}{16}=\frac{13}{16} x-\frac{7}{16}=-\frac{13}{16}$$

Add \frac{7}{16} to both sides of the equation.

$$x=\frac{5}{4} x=-\frac{3}{8}$$