Solve x^2+7x-18

$${ x }^{ 2 } +7x-18$$

Factor

$$\left(x-2\right)\left(x+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-18. To find a and b, set up a system to be solved.

$$a+b=7 ab=1\left(-18\right)=-18$$

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 -18.

$$-1,18 -2,9 -3,6$$

Calculate the sum for each pair.

$$-1+18=17 -2+9=7 -3+6=3$$

The solution is the pair that gives sum 7.

$$a=-2 b=9$$

Rewrite x^{2}+7x-18 as \left(x^{2}-2x\right)+\left(9x-18\right).

$$\left(x^{2}-2x\right)+\left(9x-18\right)$$

Factor out x in the first and 9 in the second group.

$$x\left(x-2\right)+9\left(x-2\right)$$

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

$$\left(x-2\right)\left(x+9\right)$$

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.

$$x^{2}+7x-18=0$$

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.

$$x=\frac{-7±\sqrt{7^{2}-4\left(-18\right)}}{2}$$

Square 7.

$$x=\frac{-7±\sqrt{49-4\left(-18\right)}}{2}$$

Multiply -4 times -18.

$$x=\frac{-7±\sqrt{49+72}}{2}$$

Add 49 to 72.

$$x=\frac{-7±\sqrt{121}}{2}$$

Take the square root of 121.

$$x=\frac{-7±11}{2}$$

Now solve the equation x=\frac{-7±11}{2} when ± is plus. Add -7 to 11.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-7±11}{2} when ± is minus. Subtract 11 from -7.

$$x=-\frac{18}{2}$$

Divide -18 by 2.

$$x=-9$$

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 -9 for x_{2}.

$$x^{2}+7x-18=\left(x-2\right)\left(x-\left(-9\right)\right)$$

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

$$x^{2}+7x-18=\left(x-2\right)\left(x+9\right)$$

Evaluate

$$\left(x-2\right)\left(x+9\right)$$