Solve 2x^2+14x-16

$$2 { x }^{ 2 } +14x-16$$

Factor

$$2\left(x-1\right)\left(x+8\right)$$

Solution Steps

Factor out 2.

$$2\left(x^{2}+7x-8\right)$$

Consider x^{2}+7x-8. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-8. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\left(-8\right)=-8$$

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 -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 7.

$$a=-1 b=8$$

Rewrite x^{2}+7x-8 as \left(x^{2}-x\right)+\left(8x-8\right).

$$\left(x^{2}-x\right)+\left(8x-8\right)$$

Factor out x in the first and 8 in the second group.

$$x\left(x-1\right)+8\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+8\right)$$

Rewrite the complete factored expression.

$$2\left(x-1\right)\left(x+8\right)$$

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.

$$2x^{2}+14x-16=0$$

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.

$$x=\frac{-14±\sqrt{14^{2}-4\times 2\left(-16\right)}}{2\times 2}$$

Square 14.

$$x=\frac{-14±\sqrt{196-4\times 2\left(-16\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-14±\sqrt{196-8\left(-16\right)}}{2\times 2}$$

Multiply -8 times -16.

$$x=\frac{-14±\sqrt{196+128}}{2\times 2}$$

Add 196 to 128.

$$x=\frac{-14±\sqrt{324}}{2\times 2}$$

Take the square root of 324.

$$x=\frac{-14±18}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-14±18}{4}$$

Now solve the equation x=\frac{-14±18}{4} when ± is plus. Add -14 to 18.

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

Divide 4 by 4.

$$x=1$$

Now solve the equation x=\frac{-14±18}{4} when ± is minus. Subtract 18 from -14.

$$x=-\frac{32}{4}$$

Divide -32 by 4.

$$x=-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -8 for x_{2}.

$$2x^{2}+14x-16=2\left(x-1\right)\left(x-\left(-8\right)\right)$$

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

$$2x^{2}+14x-16=2\left(x-1\right)\left(x+8\right)$$

Evaluate

$$2\left(x-1\right)\left(x+8\right)$$