Solve 2x^2+28x-960

$$2 { x }^{ 2 } +28x-960$$

Factor

$$2\left(x-16\right)\left(x+30\right)$$

Solution Steps

Factor out 2.

$$2\left(x^{2}+14x-480\right)$$

Consider x^{2}+14x-480. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-480. To find a and b, set up a system to be solved.

$$a+b=14 ab=1\left(-480\right)=-480$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -480.

$$-1,480 -2,240 -3,160 -4,120 -5,96 -6,80 -8,60 -10,48 -12,40 -15,32 -16,30 -20,24$$

Calculate the sum for each pair.

$$-1+480=479 -2+240=238 -3+160=157 -4+120=116 -5+96=91 -6+80=74 -8+60=52 -10+48=38 -12+40=28 -15+32=17 -16+30=14 -20+24=4$$

The solution is the pair that gives sum 14.

$$a=-16 b=30$$

Rewrite x^{2}+14x-480 as \left(x^{2}-16x\right)+\left(30x-480\right).

$$\left(x^{2}-16x\right)+\left(30x-480\right)$$

Factor out x in the first and 30 in the second group.

$$x\left(x-16\right)+30\left(x-16\right)$$

Factor out common term x-16 by using distributive property.

$$\left(x-16\right)\left(x+30\right)$$

Rewrite the complete factored expression.

$$2\left(x-16\right)\left(x+30\right)$$

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.

$$2x^{2}+28x-960=0$$

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.

$$x=\frac{-28±\sqrt{28^{2}-4\times 2\left(-960\right)}}{2\times 2}$$

Square 28.

$$x=\frac{-28±\sqrt{784-4\times 2\left(-960\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-28±\sqrt{784-8\left(-960\right)}}{2\times 2}$$

Multiply -8 times -960.

$$x=\frac{-28±\sqrt{784+7680}}{2\times 2}$$

Add 784 to 7680.

$$x=\frac{-28±\sqrt{8464}}{2\times 2}$$

Take the square root of 8464.

$$x=\frac{-28±92}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-28±92}{4}$$

Now solve the equation x=\frac{-28±92}{4} when ± is plus. Add -28 to 92.

$$x=\frac{64}{4}$$

Divide 64 by 4.

$$x=16$$

Now solve the equation x=\frac{-28±92}{4} when ± is minus. Subtract 92 from -28.

$$x=-\frac{120}{4}$$

Divide -120 by 4.

$$x=-30$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 16 for x_{1} and -30 for x_{2}.

$$2x^{2}+28x-960=2\left(x-16\right)\left(x-\left(-30\right)\right)$$

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

$$2x^{2}+28x-960=2\left(x-16\right)\left(x+30\right)$$

Evaluate

$$2\left(x-16\right)\left(x+30\right)$$