Solve 3x^2+33x+90=

$$3 { x }^{ 2 } +33x+90=$$

Factor

$$3\left(x+5\right)\left(x+6\right)$$

Solution Steps

Factor out 3.

$$3\left(x^{2}+11x+30\right)$$

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.

$$3\left(x+5\right)\left(x+6\right)$$

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.

$$3x^{2}+33x+90=0$$

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.

$$x=\frac{-33±\sqrt{33^{2}-4\times 3\times 90}}{2\times 3}$$

Square 33.

$$x=\frac{-33±\sqrt{1089-4\times 3\times 90}}{2\times 3}$$

Multiply -4 times 3.

$$x=\frac{-33±\sqrt{1089-12\times 90}}{2\times 3}$$

Multiply -12 times 90.

$$x=\frac{-33±\sqrt{1089-1080}}{2\times 3}$$

Add 1089 to -1080.

$$x=\frac{-33±\sqrt{9}}{2\times 3}$$

Take the square root of 9.

$$x=\frac{-33±3}{2\times 3}$$

Multiply 2 times 3.

$$x=\frac{-33±3}{6}$$

Now solve the equation x=\frac{-33±3}{6} when ± is plus. Add -33 to 3.

$$x=-\frac{30}{6}$$

Divide -30 by 6.

$$x=-5$$

Now solve the equation x=\frac{-33±3}{6} when ± is minus. Subtract 3 from -33.

$$x=-\frac{36}{6}$$

Divide -36 by 6.

$$x=-6$$

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}.

$$3x^{2}+33x+90=3\left(x-\left(-5\right)\right)\left(x-\left(-6\right)\right)$$

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

$$3x^{2}+33x+90=3\left(x+5\right)\left(x+6\right)$$

Evaluate

$$3\left(x+5\right)\left(x+6\right)$$