Solve 96x^2+44x-21=

$$96 { x }^{ 2 } +44x-21=$$

Factor

$$\left(24x-7\right)\left(4x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 96x^{2}+ax+bx-21. To find a and b, set up a system to be solved.

$$a+b=44 ab=96\left(-21\right)=-2016$$

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 -2016.

$$-1,2016 -2,1008 -3,672 -4,504 -6,336 -7,288 -8,252 -9,224 -12,168 -14,144 -16,126 -18,112 -21,96 -24,84 -28,72 -32,63 -36,56 -42,48$$

Calculate the sum for each pair.

$$-1+2016=2015 -2+1008=1006 -3+672=669 -4+504=500 -6+336=330 -7+288=281 -8+252=244 -9+224=215 -12+168=156 -14+144=130 -16+126=110 -18+112=94 -21+96=75 -24+84=60 -28+72=44 -32+63=31 -36+56=20 -42+48=6$$

The solution is the pair that gives sum 44.

$$a=-28 b=72$$

Rewrite 96x^{2}+44x-21 as \left(96x^{2}-28x\right)+\left(72x-21\right).

$$\left(96x^{2}-28x\right)+\left(72x-21\right)$$

Factor out 4x in the first and 3 in the second group.

$$4x\left(24x-7\right)+3\left(24x-7\right)$$

Factor out common term 24x-7 by using distributive property.

$$\left(24x-7\right)\left(4x+3\right)$$

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.

$$96x^{2}+44x-21=0$$

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.

$$x=\frac{-44±\sqrt{44^{2}-4\times 96\left(-21\right)}}{2\times 96}$$

Square 44.

$$x=\frac{-44±\sqrt{1936-4\times 96\left(-21\right)}}{2\times 96}$$

Multiply -4 times 96.

$$x=\frac{-44±\sqrt{1936-384\left(-21\right)}}{2\times 96}$$

Multiply -384 times -21.

$$x=\frac{-44±\sqrt{1936+8064}}{2\times 96}$$

Add 1936 to 8064.

$$x=\frac{-44±\sqrt{10000}}{2\times 96}$$

Take the square root of 10000.

$$x=\frac{-44±100}{2\times 96}$$

Multiply 2 times 96.

$$x=\frac{-44±100}{192}$$

Now solve the equation x=\frac{-44±100}{192} when ± is plus. Add -44 to 100.

$$x=\frac{56}{192}$$

Reduce the fraction \frac{56}{192} to lowest terms by extracting and canceling out 8.

$$x=\frac{7}{24}$$

Now solve the equation x=\frac{-44±100}{192} when ± is minus. Subtract 100 from -44.

$$x=-\frac{144}{192}$$

Reduce the fraction \frac{-144}{192} to lowest terms by extracting and canceling out 48.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7}{24} for x_{1} and -\frac{3}{4} for x_{2}.

$$96x^{2}+44x-21=96\left(x-\frac{7}{24}\right)\left(x-\left(-\frac{3}{4}\right)\right)$$

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

$$96x^{2}+44x-21=96\left(x-\frac{7}{24}\right)\left(x+\frac{3}{4}\right)$$

Subtract \frac{7}{24} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$96x^{2}+44x-21=96\times \frac{24x-7}{24}\left(x+\frac{3}{4}\right)$$

Add \frac{3}{4} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$96x^{2}+44x-21=96\times \frac{24x-7}{24}\times \frac{4x+3}{4}$$

Multiply \frac{24x-7}{24} times \frac{4x+3}{4} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$96x^{2}+44x-21=96\times \frac{\left(24x-7\right)\left(4x+3\right)}{24\times 4}$$

Multiply 24 times 4.

$$96x^{2}+44x-21=96\times \frac{\left(24x-7\right)\left(4x+3\right)}{96}$$

Cancel out 96, the greatest common factor in 96 and 96.

$$96x^{2}+44x-21=\left(24x-7\right)\left(4x+3\right)$$

Evaluate

$$\left(24x-7\right)\left(4x+3\right)$$