Solve x^2-15x-100

$${ x }^{ 2 } -15x-100$$

Factor

$$\left(x-20\right)\left(x+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-100. To find a and b, set up a system to be solved.

$$a+b=-15 ab=1\left(-100\right)=-100$$

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 -100.

$$1,-100 2,-50 4,-25 5,-20 10,-10$$

Calculate the sum for each pair.

$$1-100=-99 2-50=-48 4-25=-21 5-20=-15 10-10=0$$

The solution is the pair that gives sum -15.

$$a=-20 b=5$$

Rewrite x^{2}-15x-100 as \left(x^{2}-20x\right)+\left(5x-100\right).

$$\left(x^{2}-20x\right)+\left(5x-100\right)$$

Factor out x in the first and 5 in the second group.

$$x\left(x-20\right)+5\left(x-20\right)$$

Factor out common term x-20 by using distributive property.

$$\left(x-20\right)\left(x+5\right)$$

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.

$$x^{2}-15x-100=0$$

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.

$$x=\frac{-\left(-15\right)±\sqrt{\left(-15\right)^{2}-4\left(-100\right)}}{2}$$

Square -15.

$$x=\frac{-\left(-15\right)±\sqrt{225-4\left(-100\right)}}{2}$$

Multiply -4 times -100.

$$x=\frac{-\left(-15\right)±\sqrt{225+400}}{2}$$

Add 225 to 400.

$$x=\frac{-\left(-15\right)±\sqrt{625}}{2}$$

Take the square root of 625.

$$x=\frac{-\left(-15\right)±25}{2}$$

The opposite of -15 is 15.

$$x=\frac{15±25}{2}$$

Now solve the equation x=\frac{15±25}{2} when ± is plus. Add 15 to 25.

$$x=\frac{40}{2}$$

Divide 40 by 2.

$$x=20$$

Now solve the equation x=\frac{15±25}{2} when ± is minus. Subtract 25 from 15.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 20 for x_{1} and -5 for x_{2}.

$$x^{2}-15x-100=\left(x-20\right)\left(x-\left(-5\right)\right)$$

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

$$x^{2}-15x-100=\left(x-20\right)\left(x+5\right)$$

Evaluate

$$\left(x-20\right)\left(x+5\right)$$