Solve 4a^2+4a-80

$$4 { a }^{ 2 } +4a-80$$

Factor

$$4\left(a-4\right)\left(a+5\right)$$

Solution Steps

Factor out 4.

$$4\left(a^{2}+a-20\right)$$

Consider a^{2}+a-20. Factor the expression by grouping. First, the expression needs to be rewritten as a^{2}+pa+qa-20. To find p and q, set up a system to be solved.

$$p+q=1 pq=1\left(-20\right)=-20$$

Since pq is negative, p and q have the opposite signs. Since p+q is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -20.

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

Calculate the sum for each pair.

$$-1+20=19 -2+10=8 -4+5=1$$

The solution is the pair that gives sum 1.

$$p=-4 q=5$$

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

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

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

$$a\left(a-4\right)+5\left(a-4\right)$$

Factor out common term a-4 by using distributive property.

$$\left(a-4\right)\left(a+5\right)$$

Rewrite the complete factored expression.

$$4\left(a-4\right)\left(a+5\right)$$

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.

$$4a^{2}+4a-80=0$$

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.

$$a=\frac{-4±\sqrt{4^{2}-4\times 4\left(-80\right)}}{2\times 4}$$

Square 4.

$$a=\frac{-4±\sqrt{16-4\times 4\left(-80\right)}}{2\times 4}$$

Multiply -4 times 4.

$$a=\frac{-4±\sqrt{16-16\left(-80\right)}}{2\times 4}$$

Multiply -16 times -80.

$$a=\frac{-4±\sqrt{16+1280}}{2\times 4}$$

Add 16 to 1280.

$$a=\frac{-4±\sqrt{1296}}{2\times 4}$$

Take the square root of 1296.

$$a=\frac{-4±36}{2\times 4}$$

Multiply 2 times 4.

$$a=\frac{-4±36}{8}$$

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

$$a=\frac{32}{8}$$

Divide 32 by 8.

$$a=4$$

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

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

Divide -40 by 8.

$$a=-5$$

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}.

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

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

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

Evaluate

$$4\left(a-4\right)\left(a+5\right)$$