Solve x^2-13x-48

$${ x }^{ 2 } -13x-48$$

Factor

$$\left(x-16\right)\left(x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-48. To find a and b, set up a system to be solved.

$$a+b=-13 ab=1\left(-48\right)=-48$$

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 -48.

$$1,-48 2,-24 3,-16 4,-12 6,-8$$

Calculate the sum for each pair.

$$1-48=-47 2-24=-22 3-16=-13 4-12=-8 6-8=-2$$

The solution is the pair that gives sum -13.

$$a=-16 b=3$$

Rewrite x^{2}-13x-48 as \left(x^{2}-16x\right)+\left(3x-48\right).

$$\left(x^{2}-16x\right)+\left(3x-48\right)$$

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

$$x\left(x-16\right)+3\left(x-16\right)$$

Factor out common term x-16 by using distributive property.

$$\left(x-16\right)\left(x+3\right)$$

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.

$$x^{2}-13x-48=0$$

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.

$$x=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-48\right)}}{2}$$

Square -13.

$$x=\frac{-\left(-13\right)±\sqrt{169-4\left(-48\right)}}{2}$$

Multiply -4 times -48.

$$x=\frac{-\left(-13\right)±\sqrt{169+192}}{2}$$

Add 169 to 192.

$$x=\frac{-\left(-13\right)±\sqrt{361}}{2}$$

Take the square root of 361.

$$x=\frac{-\left(-13\right)±19}{2}$$

The opposite of -13 is 13.

$$x=\frac{13±19}{2}$$

Now solve the equation x=\frac{13±19}{2} when ± is plus. Add 13 to 19.

$$x=\frac{32}{2}$$

Divide 32 by 2.

$$x=16$$

Now solve the equation x=\frac{13±19}{2} when ± is minus. Subtract 19 from 13.

$$x=-\frac{6}{2}$$

Divide -6 by 2.

$$x=-3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 16 for x_{1} and -3 for x_{2}.

$$x^{2}-13x-48=\left(x-16\right)\left(x-\left(-3\right)\right)$$

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

$$x^{2}-13x-48=\left(x-16\right)\left(x+3\right)$$

Evaluate

$$\left(x-16\right)\left(x+3\right)$$