Solve 32x^2-4x-6

$$32 { x }^{ 2 } -4x-6$$

Factor

$$2\left(2x-1\right)\left(8x+3\right)$$

Solution Steps

Factor out 2.

$$2\left(16x^{2}-2x-3\right)$$

Consider 16x^{2}-2x-3. Factor the expression by grouping. First, the expression needs to be rewritten as 16x^{2}+ax+bx-3. To find a and b, set up a system to be solved.

$$a+b=-2 ab=16\left(-3\right)=-48$$

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 -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -2.

$$a=-8 b=6$$

Rewrite 16x^{2}-2x-3 as \left(16x^{2}-8x\right)+\left(6x-3\right).

$$\left(16x^{2}-8x\right)+\left(6x-3\right)$$

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

$$8x\left(2x-1\right)+3\left(2x-1\right)$$

Factor out common term 2x-1 by using distributive property.

$$\left(2x-1\right)\left(8x+3\right)$$

Rewrite the complete factored expression.

$$2\left(2x-1\right)\left(8x+3\right)$$

Steps Using the Quadratic Formula

Quadratic polynomial can be factored using the transformation ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right), where x_{1} and x_{2} are the solutions of the quadratic equation ax^{2}+bx+c=0.

$$32x^{2}-4x-6=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$x=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 32\left(-6\right)}}{2\times 32}$$

Square -4.

$$x=\frac{-\left(-4\right)±\sqrt{16-4\times 32\left(-6\right)}}{2\times 32}$$

Multiply -4 times 32.

$$x=\frac{-\left(-4\right)±\sqrt{16-128\left(-6\right)}}{2\times 32}$$

Multiply -128 times -6.

$$x=\frac{-\left(-4\right)±\sqrt{16+768}}{2\times 32}$$

Add 16 to 768.

$$x=\frac{-\left(-4\right)±\sqrt{784}}{2\times 32}$$

Take the square root of 784.

$$x=\frac{-\left(-4\right)±28}{2\times 32}$$

The opposite of -4 is 4.

$$x=\frac{4±28}{2\times 32}$$

Multiply 2 times 32.

$$x=\frac{4±28}{64}$$

Now solve the equation x=\frac{4±28}{64} when ± is plus. Add 4 to 28.

$$x=\frac{32}{64}$$

Reduce the fraction \frac{32}{64} to lowest terms by extracting and canceling out 32.

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

Now solve the equation x=\frac{4±28}{64} when ± is minus. Subtract 28 from 4.

$$x=-\frac{24}{64}$$

Reduce the fraction \frac{-24}{64} to lowest terms by extracting and canceling out 8.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{2} for x_{1} and -\frac{3}{8} for x_{2}.

$$32x^{2}-4x-6=32\left(x-\frac{1}{2}\right)\left(x-\left(-\frac{3}{8}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$32x^{2}-4x-6=32\left(x-\frac{1}{2}\right)\left(x+\frac{3}{8}\right)$$

Subtract \frac{1}{2} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$32x^{2}-4x-6=32\times \frac{2x-1}{2}\left(x+\frac{3}{8}\right)$$

Add \frac{3}{8} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$32x^{2}-4x-6=32\times \frac{2x-1}{2}\times \frac{8x+3}{8}$$

Multiply \frac{2x-1}{2} times \frac{8x+3}{8} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$32x^{2}-4x-6=32\times \frac{\left(2x-1\right)\left(8x+3\right)}{2\times 8}$$

Multiply 2 times 8.

$$32x^{2}-4x-6=32\times \frac{\left(2x-1\right)\left(8x+3\right)}{16}$$

Cancel out 16, the greatest common factor in 32 and 16.

$$32x^{2}-4x-6=2\left(2x-1\right)\left(8x+3\right)$$

Evaluate

$$32x^{2}-4x-6$$