Solve x^2-3x-238

$${ x }^{ 2 } -3x-238$$

Factor

$$\left(x-17\right)\left(x+14\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-238. To find a and b, set up a system to be solved.

$$a+b=-3 ab=1\left(-238\right)=-238$$

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 -238.

$$1,-238 2,-119 7,-34 14,-17$$

Calculate the sum for each pair.

$$1-238=-237 2-119=-117 7-34=-27 14-17=-3$$

The solution is the pair that gives sum -3.

$$a=-17 b=14$$

Rewrite x^{2}-3x-238 as \left(x^{2}-17x\right)+\left(14x-238\right).

$$\left(x^{2}-17x\right)+\left(14x-238\right)$$

Factor out x in the first and 14 in the second group.

$$x\left(x-17\right)+14\left(x-17\right)$$

Factor out common term x-17 by using distributive property.

$$\left(x-17\right)\left(x+14\right)$$

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.

$$x^{2}-3x-238=0$$

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.

$$x=\frac{-\left(-3\right)±\sqrt{\left(-3\right)^{2}-4\left(-238\right)}}{2}$$

Square -3.

$$x=\frac{-\left(-3\right)±\sqrt{9-4\left(-238\right)}}{2}$$

Multiply -4 times -238.

$$x=\frac{-\left(-3\right)±\sqrt{9+952}}{2}$$

Add 9 to 952.

$$x=\frac{-\left(-3\right)±\sqrt{961}}{2}$$

Take the square root of 961.

$$x=\frac{-\left(-3\right)±31}{2}$$

The opposite of -3 is 3.

$$x=\frac{3±31}{2}$$

Now solve the equation x=\frac{3±31}{2} when ± is plus. Add 3 to 31.

$$x=\frac{34}{2}$$

Divide 34 by 2.

$$x=17$$

Now solve the equation x=\frac{3±31}{2} when ± is minus. Subtract 31 from 3.

$$x=-\frac{28}{2}$$

Divide -28 by 2.

$$x=-14$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 17 for x_{1} and -14 for x_{2}.

$$x^{2}-3x-238=\left(x-17\right)\left(x-\left(-14\right)\right)$$

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

$$x^{2}-3x-238=\left(x-17\right)\left(x+14\right)$$

Evaluate

$$\left(x-17\right)\left(x+14\right)$$