Solve 48x^2+8x-1

$$48 { x }^{ 2 } +8x-1$$

Factor

$$\left(12x-1\right)\left(4x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 48x^{2}+ax+bx-1. To find a and b, set up a system to be solved.

$$a+b=8 ab=48\left(-1\right)=-48$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. 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 8.

$$a=-4 b=12$$

Rewrite 48x^{2}+8x-1 as \left(48x^{2}-4x\right)+\left(12x-1\right).

$$\left(48x^{2}-4x\right)+\left(12x-1\right)$$

Factor out 4x in 48x^{2}-4x.

$$4x\left(12x-1\right)+12x-1$$

Factor out common term 12x-1 by using distributive property.

$$\left(12x-1\right)\left(4x+1\right)$$

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.

$$48x^{2}+8x-1=0$$

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.

$$x=\frac{-8±\sqrt{8^{2}-4\times 48\left(-1\right)}}{2\times 48}$$

Square 8.

$$x=\frac{-8±\sqrt{64-4\times 48\left(-1\right)}}{2\times 48}$$

Multiply -4 times 48.

$$x=\frac{-8±\sqrt{64-192\left(-1\right)}}{2\times 48}$$

Multiply -192 times -1.

$$x=\frac{-8±\sqrt{64+192}}{2\times 48}$$

Add 64 to 192.

$$x=\frac{-8±\sqrt{256}}{2\times 48}$$

Take the square root of 256.

$$x=\frac{-8±16}{2\times 48}$$

Multiply 2 times 48.

$$x=\frac{-8±16}{96}$$

Now solve the equation x=\frac{-8±16}{96} when ± is plus. Add -8 to 16.

$$x=\frac{8}{96}$$

Reduce the fraction \frac{8}{96} to lowest terms by extracting and canceling out 8.

$$x=\frac{1}{12}$$

Now solve the equation x=\frac{-8±16}{96} when ± is minus. Subtract 16 from -8.

$$x=-\frac{24}{96}$$

Reduce the fraction \frac{-24}{96} to lowest terms by extracting and canceling out 24.

$$x=-\frac{1}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{1}{12} for x_{1} and -\frac{1}{4} for x_{2}.

$$48x^{2}+8x-1=48\left(x-\frac{1}{12}\right)\left(x-\left(-\frac{1}{4}\right)\right)$$

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

$$48x^{2}+8x-1=48\left(x-\frac{1}{12}\right)\left(x+\frac{1}{4}\right)$$

Subtract \frac{1}{12} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$48x^{2}+8x-1=48\times \frac{12x-1}{12}\left(x+\frac{1}{4}\right)$$

Add \frac{1}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$48x^{2}+8x-1=48\times \frac{12x-1}{12}\times \frac{4x+1}{4}$$

Multiply \frac{12x-1}{12} times \frac{4x+1}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$48x^{2}+8x-1=48\times \frac{\left(12x-1\right)\left(4x+1\right)}{12\times 4}$$

Multiply 12 times 4.

$$48x^{2}+8x-1=48\times \frac{\left(12x-1\right)\left(4x+1\right)}{48}$$

Cancel out 48, the greatest common factor in 48 and 48.

$$48x^{2}+8x-1=\left(12x-1\right)\left(4x+1\right)$$

Evaluate

$$\left(12x-1\right)\left(4x+1\right)$$