Solve x^2+x-72=

$${ x }^{ 2 } +x-72=$$

Factor

$$\left(x-8\right)\left(x+9\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-72. To find a and b, set up a system to be solved.

$$a+b=1 ab=1\left(-72\right)=-72$$

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 -72.

$$-1,72 -2,36 -3,24 -4,18 -6,12 -8,9$$

Calculate the sum for each pair.

$$-1+72=71 -2+36=34 -3+24=21 -4+18=14 -6+12=6 -8+9=1$$

The solution is the pair that gives sum 1.

$$a=-8 b=9$$

Rewrite x^{2}+x-72 as \left(x^{2}-8x\right)+\left(9x-72\right).

$$\left(x^{2}-8x\right)+\left(9x-72\right)$$

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

$$x\left(x-8\right)+9\left(x-8\right)$$

Factor out common term x-8 by using distributive property.

$$\left(x-8\right)\left(x+9\right)$$

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.

$$x^{2}+x-72=0$$

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.

$$x=\frac{-1±\sqrt{1^{2}-4\left(-72\right)}}{2}$$

Square 1.

$$x=\frac{-1±\sqrt{1-4\left(-72\right)}}{2}$$

Multiply -4 times -72.

$$x=\frac{-1±\sqrt{1+288}}{2}$$

Add 1 to 288.

$$x=\frac{-1±\sqrt{289}}{2}$$

Take the square root of 289.

$$x=\frac{-1±17}{2}$$

Now solve the equation x=\frac{-1±17}{2} when ± is plus. Add -1 to 17.

$$x=\frac{16}{2}$$

Divide 16 by 2.

$$x=8$$

Now solve the equation x=\frac{-1±17}{2} when ± is minus. Subtract 17 from -1.

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

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 8 for x_{1} and -9 for x_{2}.

$$x^{2}+x-72=\left(x-8\right)\left(x-\left(-9\right)\right)$$

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

$$x^{2}+x-72=\left(x-8\right)\left(x+9\right)$$

Evaluate

$$\left(x-8\right)\left(x+9\right)$$