Solve 2x^2+4x-30

$$2 { x }^{ 2 } +4x-30$$

Factor

$$2\left(x-3\right)\left(x+5\right)$$

Solution Steps

Factor out 2.

$$2\left(x^{2}+2x-15\right)$$

Consider x^{2}+2x-15. Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-15. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-15\right)=-15$$

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 -15.

$$-1,15 -3,5$$

Calculate the sum for each pair.

$$-1+15=14 -3+5=2$$

The solution is the pair that gives sum 2.

$$a=-3 b=5$$

Rewrite x^{2}+2x-15 as \left(x^{2}-3x\right)+\left(5x-15\right).

$$\left(x^{2}-3x\right)+\left(5x-15\right)$$

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

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

Factor out common term x-3 by using distributive property.

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

Rewrite the complete factored expression.

$$2\left(x-3\right)\left(x+5\right)$$

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.

$$2x^{2}+4x-30=0$$

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.

$$x=\frac{-4±\sqrt{4^{2}-4\times 2\left(-30\right)}}{2\times 2}$$

Square 4.

$$x=\frac{-4±\sqrt{16-4\times 2\left(-30\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-4±\sqrt{16-8\left(-30\right)}}{2\times 2}$$

Multiply -8 times -30.

$$x=\frac{-4±\sqrt{16+240}}{2\times 2}$$

Add 16 to 240.

$$x=\frac{-4±\sqrt{256}}{2\times 2}$$

Take the square root of 256.

$$x=\frac{-4±16}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-4±16}{4}$$

Now solve the equation x=\frac{-4±16}{4} when ± is plus. Add -4 to 16.

$$x=\frac{12}{4}$$

Divide 12 by 4.

$$x=3$$

Now solve the equation x=\frac{-4±16}{4} when ± is minus. Subtract 16 from -4.

$$x=-\frac{20}{4}$$

Divide -20 by 4.

$$x=-5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3 for x_{1} and -5 for x_{2}.

$$2x^{2}+4x-30=2\left(x-3\right)\left(x-\left(-5\right)\right)$$

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

$$2x^{2}+4x-30=2\left(x-3\right)\left(x+5\right)$$

Evaluate

$$2\left(x-3\right)\left(x+5\right)$$