Solve 2x^2+11x-30

$$2 { x }^{ 2 } +11x-30$$

Factor

$$\left(x-2\right)\left(2x+15\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 2x^{2}+ax+bx-30. To find a and b, set up a system to be solved.

$$a+b=11 ab=2\left(-30\right)=-60$$

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 -60.

$$-1,60 -2,30 -3,20 -4,15 -5,12 -6,10$$

Calculate the sum for each pair.

$$-1+60=59 -2+30=28 -3+20=17 -4+15=11 -5+12=7 -6+10=4$$

The solution is the pair that gives sum 11.

$$a=-4 b=15$$

Rewrite 2x^{2}+11x-30 as \left(2x^{2}-4x\right)+\left(15x-30\right).

$$\left(2x^{2}-4x\right)+\left(15x-30\right)$$

Factor out 2x in the first and 15 in the second group.

$$2x\left(x-2\right)+15\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(2x+15\right)$$

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.

$$2x^{2}+11x-30=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\times 2\left(-30\right)}}{2\times 2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\times 2\left(-30\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-11±\sqrt{121-8\left(-30\right)}}{2\times 2}$$

Multiply -8 times -30.

$$x=\frac{-11±\sqrt{121+240}}{2\times 2}$$

Add 121 to 240.

$$x=\frac{-11±\sqrt{361}}{2\times 2}$$

Take the square root of 361.

$$x=\frac{-11±19}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{-11±19}{4}$$

Now solve the equation x=\frac{-11±19}{4} when ± is plus. Add -11 to 19.

$$x=\frac{8}{4}$$

Divide 8 by 4.

$$x=2$$

Now solve the equation x=\frac{-11±19}{4} when ± is minus. Subtract 19 from -11.

$$x=-\frac{30}{4}$$

Reduce the fraction \frac{-30}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{15}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -\frac{15}{2} for x_{2}.

$$2x^{2}+11x-30=2\left(x-2\right)\left(x-\left(-\frac{15}{2}\right)\right)$$

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

$$2x^{2}+11x-30=2\left(x-2\right)\left(x+\frac{15}{2}\right)$$

Add \frac{15}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2x^{2}+11x-30=2\left(x-2\right)\times \frac{2x+15}{2}$$

Cancel out 2, the greatest common factor in 2 and 2.

$$2x^{2}+11x-30=\left(x-2\right)\left(2x+15\right)$$

Evaluate

$$\left(x-2\right)\left(2x+15\right)$$