Solve 12n^2+58n+56

$$12 { n }^{ 2 } +58n+56$$

Factor

$$2\left(2n+7\right)\left(3n+4\right)$$

Solution Steps

Factor out 2.

$$2\left(6n^{2}+29n+28\right)$$

Consider 6n^{2}+29n+28. Factor the expression by grouping. First, the expression needs to be rewritten as 6n^{2}+an+bn+28. To find a and b, set up a system to be solved.

$$a+b=29 ab=6\times 28=168$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 168.

$$1,168 2,84 3,56 4,42 6,28 7,24 8,21 12,14$$

Calculate the sum for each pair.

$$1+168=169 2+84=86 3+56=59 4+42=46 6+28=34 7+24=31 8+21=29 12+14=26$$

The solution is the pair that gives sum 29.

$$a=8 b=21$$

Rewrite 6n^{2}+29n+28 as \left(6n^{2}+8n\right)+\left(21n+28\right).

$$\left(6n^{2}+8n\right)+\left(21n+28\right)$$

Factor out 2n in the first and 7 in the second group.

$$2n\left(3n+4\right)+7\left(3n+4\right)$$

Factor out common term 3n+4 by using distributive property.

$$\left(3n+4\right)\left(2n+7\right)$$

Rewrite the complete factored expression.

$$2\left(3n+4\right)\left(2n+7\right)$$

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.

$$12n^{2}+58n+56=0$$

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.

$$n=\frac{-58±\sqrt{58^{2}-4\times 12\times 56}}{2\times 12}$$

Square 58.

$$n=\frac{-58±\sqrt{3364-4\times 12\times 56}}{2\times 12}$$

Multiply -4 times 12.

$$n=\frac{-58±\sqrt{3364-48\times 56}}{2\times 12}$$

Multiply -48 times 56.

$$n=\frac{-58±\sqrt{3364-2688}}{2\times 12}$$

Add 3364 to -2688.

$$n=\frac{-58±\sqrt{676}}{2\times 12}$$

Take the square root of 676.

$$n=\frac{-58±26}{2\times 12}$$

Multiply 2 times 12.

$$n=\frac{-58±26}{24}$$

Now solve the equation n=\frac{-58±26}{24} when ± is plus. Add -58 to 26.

$$n=-\frac{32}{24}$$

Reduce the fraction \frac{-32}{24} to lowest terms by extracting and canceling out 8.

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

Now solve the equation n=\frac{-58±26}{24} when ± is minus. Subtract 26 from -58.

$$n=-\frac{84}{24}$$

Reduce the fraction \frac{-84}{24} to lowest terms by extracting and canceling out 12.

$$n=-\frac{7}{2}$$

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

$$12n^{2}+58n+56=12\left(n-\left(-\frac{4}{3}\right)\right)\left(n-\left(-\frac{7}{2}\right)\right)$$

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

$$12n^{2}+58n+56=12\left(n+\frac{4}{3}\right)\left(n+\frac{7}{2}\right)$$

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

$$12n^{2}+58n+56=12\times \frac{3n+4}{3}\left(n+\frac{7}{2}\right)$$

Add \frac{7}{2} to n by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$12n^{2}+58n+56=12\times \frac{3n+4}{3}\times \frac{2n+7}{2}$$

Multiply \frac{3n+4}{3} times \frac{2n+7}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$12n^{2}+58n+56=12\times \frac{\left(3n+4\right)\left(2n+7\right)}{3\times 2}$$

Multiply 3 times 2.

$$12n^{2}+58n+56=12\times \frac{\left(3n+4\right)\left(2n+7\right)}{6}$$

Cancel out 6, the greatest common factor in 12 and 6.

$$12n^{2}+58n+56=2\left(3n+4\right)\left(2n+7\right)$$

Evaluate

$$12n^{2}+58n+56$$