Solve 2x^2+22x+60

$$2 { x }^{ 2 } +22x+60$$

Factor

$$2\left(x+5\right)\left(x+6\right)$$

Solution Steps

Factor out 2.

$$2\left(x^{2}+11x+30\right)$$

Consider x^{2}+11x+30. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+30. To find a and b, set up a system to be solved.

$$a+b=11 ab=1\times 30=30$$

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 11.

$$a=5 b=6$$

Rewrite x^{2}+11x+30 as \left(x^{2}+5x\right)+\left(6x+30\right).

$$\left(x^{2}+5x\right)+\left(6x+30\right)$$

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

$$x\left(x+5\right)+6\left(x+5\right)$$

Factor out common term x+5 by using distributive property.

$$\left(x+5\right)\left(x+6\right)$$

Rewrite the complete factored expression.

$$2\left(x+5\right)\left(x+6\right)$$

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.

$$2x^{2}+22x+60=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\times 2\times 60}}{2\times 2}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 2\times 60}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-22±\sqrt{484-8\times 60}}{2\times 2}$$

Multiply -8 times 60.

$$x=\frac{-22±\sqrt{484-480}}{2\times 2}$$

Add 484 to -480.

$$x=\frac{-22±\sqrt{4}}{2\times 2}$$

Take the square root of 4.

$$x=\frac{-22±2}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-22±2}{4}$$

Now solve the equation x=\frac{-22±2}{4} when ± is plus. Add -22 to 2.

$$x=-\frac{20}{4}$$

Divide -20 by 4.

$$x=-5$$

Now solve the equation x=\frac{-22±2}{4} when ± is minus. Subtract 2 from -22.

$$x=-\frac{24}{4}$$

Divide -24 by 4.

$$x=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and -6 for x_{2}.

$$2x^{2}+22x+60=2\left(x-\left(-5\right)\right)\left(x-\left(-6\right)\right)$$

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

$$2x^{2}+22x+60=2\left(x+5\right)\left(x+6\right)$$

Evaluate

$$2\left(x+5\right)\left(x+6\right)$$