Solve 6x^2+13x+5

$$6 { x }^{ 2 } +13x+5$$

Factor

$$\left(2x+1\right)\left(3x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6x^{2}+ax+bx+5. To find a and b, set up a system to be solved.

$$a+b=13 ab=6\times 5=30$$

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 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 13.

$$a=3 b=10$$

Rewrite 6x^{2}+13x+5 as \left(6x^{2}+3x\right)+\left(10x+5\right).

$$\left(6x^{2}+3x\right)+\left(10x+5\right)$$

Factor out 3x in the first and 5 in the second group.

$$3x\left(2x+1\right)+5\left(2x+1\right)$$

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

$$\left(2x+1\right)\left(3x+5\right)$$

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.

$$6x^{2}+13x+5=0$$

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.

$$x=\frac{-13±\sqrt{13^{2}-4\times 6\times 5}}{2\times 6}$$

Square 13.

$$x=\frac{-13±\sqrt{169-4\times 6\times 5}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-13±\sqrt{169-24\times 5}}{2\times 6}$$

Multiply -24 times 5.

$$x=\frac{-13±\sqrt{169-120}}{2\times 6}$$

Add 169 to -120.

$$x=\frac{-13±\sqrt{49}}{2\times 6}$$

Take the square root of 49.

$$x=\frac{-13±7}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{-13±7}{12}$$

Now solve the equation x=\frac{-13±7}{12} when ± is plus. Add -13 to 7.

$$x=-\frac{6}{12}$$

Reduce the fraction \frac{-6}{12} to lowest terms by extracting and canceling out 6.

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

Now solve the equation x=\frac{-13±7}{12} when ± is minus. Subtract 7 from -13.

$$x=-\frac{20}{12}$$

Reduce the fraction \frac{-20}{12} to lowest terms by extracting and canceling out 4.

$$x=-\frac{5}{3}$$

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{5}{3} for x_{2}.

$$6x^{2}+13x+5=6\left(x-\left(-\frac{1}{2}\right)\right)\left(x-\left(-\frac{5}{3}\right)\right)$$

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

$$6x^{2}+13x+5=6\left(x+\frac{1}{2}\right)\left(x+\frac{5}{3}\right)$$

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

$$6x^{2}+13x+5=6\times \frac{2x+1}{2}\left(x+\frac{5}{3}\right)$$

Add \frac{5}{3} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6x^{2}+13x+5=6\times \frac{2x+1}{2}\times \frac{3x+5}{3}$$

Multiply \frac{2x+1}{2} times \frac{3x+5}{3} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$6x^{2}+13x+5=6\times \frac{\left(2x+1\right)\left(3x+5\right)}{2\times 3}$$

Multiply 2 times 3.

$$6x^{2}+13x+5=6\times \frac{\left(2x+1\right)\left(3x+5\right)}{6}$$

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

$$6x^{2}+13x+5=\left(2x+1\right)\left(3x+5\right)$$

Evaluate

$$\left(2x+1\right)\left(3x+5\right)$$