Solve 4x^2+36x+80

$$4 { x }^{ 2 } +36x+80$$

Factor

$$4\left(x+4\right)\left(x+5\right)$$

Solution Steps

Factor out 4.

$$4\left(x^{2}+9x+20\right)$$

Consider x^{2}+9x+20. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+20. To find a and b, set up a system to be solved.

$$a+b=9 ab=1\times 20=20$$

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 20.

$$1,20 2,10 4,5$$

Calculate the sum for each pair.

$$1+20=21 2+10=12 4+5=9$$

The solution is the pair that gives sum 9.

$$a=4 b=5$$

Rewrite x^{2}+9x+20 as \left(x^{2}+4x\right)+\left(5x+20\right).

$$\left(x^{2}+4x\right)+\left(5x+20\right)$$

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

$$x\left(x+4\right)+5\left(x+4\right)$$

Factor out common term x+4 by using distributive property.

$$\left(x+4\right)\left(x+5\right)$$

Rewrite the complete factored expression.

$$4\left(x+4\right)\left(x+5\right)$$

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.

$$4x^{2}+36x+80=0$$

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.

$$x=\frac{-36±\sqrt{36^{2}-4\times 4\times 80}}{2\times 4}$$

Square 36.

$$x=\frac{-36±\sqrt{1296-4\times 4\times 80}}{2\times 4}$$

Multiply -4 times 4.

$$x=\frac{-36±\sqrt{1296-16\times 80}}{2\times 4}$$

Multiply -16 times 80.

$$x=\frac{-36±\sqrt{1296-1280}}{2\times 4}$$

Add 1296 to -1280.

$$x=\frac{-36±\sqrt{16}}{2\times 4}$$

Take the square root of 16.

$$x=\frac{-36±4}{2\times 4}$$

Multiply 2 times 4.

$$x=\frac{-36±4}{8}$$

Now solve the equation x=\frac{-36±4}{8} when ± is plus. Add -36 to 4.

$$x=-\frac{32}{8}$$

Divide -32 by 8.

$$x=-4$$

Now solve the equation x=\frac{-36±4}{8} when ± is minus. Subtract 4 from -36.

$$x=-\frac{40}{8}$$

Divide -40 by 8.

$$x=-5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -4 for x_{1} and -5 for x_{2}.

$$4x^{2}+36x+80=4\left(x-\left(-4\right)\right)\left(x-\left(-5\right)\right)$$

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

$$4x^{2}+36x+80=4\left(x+4\right)\left(x+5\right)$$

Evaluate

$$4\left(x+4\right)\left(x+5\right)$$