Solve 6x^2+37x+6

$$6 { x }^{ 2 } +37x+6$$

Factor

$$\left(x+6\right)\left(6x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx+6. To find a and b, set up a system to be solved.

$$a+b=37 ab=6\times 6=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 37.

$$a=1 b=36$$

Rewrite 6x^{2}+37x+6 as \left(6x^{2}+x\right)+\left(36x+6\right).

$$\left(6x^{2}+x\right)+\left(36x+6\right)$$

Factor out x in the first and 6 in the second group.

$$x\left(6x+1\right)+6\left(6x+1\right)$$

Factor out common term 6x+1 by using distributive property.

$$\left(6x+1\right)\left(x+6\right)$$

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.

$$6x^{2}+37x+6=0$$

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.

$$x=\frac{-37±\sqrt{37^{2}-4\times 6\times 6}}{2\times 6}$$

Square 37.

$$x=\frac{-37±\sqrt{1369-4\times 6\times 6}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-37±\sqrt{1369-24\times 6}}{2\times 6}$$

Multiply -24 times 6.

$$x=\frac{-37±\sqrt{1369-144}}{2\times 6}$$

Add 1369 to -144.

$$x=\frac{-37±\sqrt{1225}}{2\times 6}$$

Take the square root of 1225.

$$x=\frac{-37±35}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-37±35}{12}$$

Now solve the equation x=\frac{-37±35}{12} when ± is plus. Add -37 to 35.

$$x=-\frac{2}{12}$$

Reduce the fraction \frac{-2}{12} to lowest terms by extracting and canceling out 2.

$$x=-\frac{1}{6}$$

Now solve the equation x=\frac{-37±35}{12} when ± is minus. Subtract 35 from -37.

$$x=-\frac{72}{12}$$

Divide -72 by 12.

$$x=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{6} for x_{1} and -6 for x_{2}.

$$6x^{2}+37x+6=6\left(x-\left(-\frac{1}{6}\right)\right)\left(x-\left(-6\right)\right)$$

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

$$6x^{2}+37x+6=6\left(x+\frac{1}{6}\right)\left(x+6\right)$$

Add \frac{1}{6} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6x^{2}+37x+6=6\times \frac{6x+1}{6}\left(x+6\right)$$

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

$$6x^{2}+37x+6=\left(6x+1\right)\left(x+6\right)$$

Evaluate

$$\left(x+6\right)\left(6x+1\right)$$