Solve 2x^2-17x-30

$$2 { x }^{ 2 } -17x-30$$

Factor

$$\left(x-10\right)\left(2x+3\right)$$

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.

$$a+b=-17 ab=2\left(-30\right)=-60$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. 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 -17.

$$a=-20 b=3$$

Rewrite 2x^{2}-17x-30 as \left(2x^{2}-20x\right)+\left(3x-30\right).

$$\left(2x^{2}-20x\right)+\left(3x-30\right)$$

Factor out 2x in the first and 3 in the second group.

$$2x\left(x-10\right)+3\left(x-10\right)$$

Factor out common term x-10 by using distributive property.

$$\left(x-10\right)\left(2x+3\right)$$

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.

$$2x^{2}-17x-30=0$$

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.

$$x=\frac{-\left(-17\right)±\sqrt{\left(-17\right)^{2}-4\times 2\left(-30\right)}}{2\times 2}$$

Square -17.

$$x=\frac{-\left(-17\right)±\sqrt{289-4\times 2\left(-30\right)}}{2\times 2}$$

Multiply -4 times 2.

$$x=\frac{-\left(-17\right)±\sqrt{289-8\left(-30\right)}}{2\times 2}$$

Multiply -8 times -30.

$$x=\frac{-\left(-17\right)±\sqrt{289+240}}{2\times 2}$$

Add 289 to 240.

$$x=\frac{-\left(-17\right)±\sqrt{529}}{2\times 2}$$

Take the square root of 529.

$$x=\frac{-\left(-17\right)±23}{2\times 2}$$

The opposite of -17 is 17.

$$x=\frac{17±23}{2\times 2}$$

Multiply 2 times 2.

$$x=\frac{17±23}{4}$$

Now solve the equation x=\frac{17±23}{4} when ± is plus. Add 17 to 23.

$$x=\frac{40}{4}$$

Divide 40 by 4.

$$x=10$$

Now solve the equation x=\frac{17±23}{4} when ± is minus. Subtract 23 from 17.

$$x=-\frac{6}{4}$$

Reduce the fraction \frac{-6}{4} to lowest terms by extracting and canceling out 2.

$$x=-\frac{3}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 10 for x_{1} and -\frac{3}{2} for x_{2}.

$$2x^{2}-17x-30=2\left(x-10\right)\left(x-\left(-\frac{3}{2}\right)\right)$$

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

$$2x^{2}-17x-30=2\left(x-10\right)\left(x+\frac{3}{2}\right)$$

Add \frac{3}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$2x^{2}-17x-30=2\left(x-10\right)\times \frac{2x+3}{2}$$

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

$$2x^{2}-17x-30=\left(x-10\right)\left(2x+3\right)$$

Evaluate

$$\left(x-10\right)\left(2x+3\right)$$