Solve x^2+23x-24

$${ x }^{ 2 } +23x-24$$

Factor

$$\left(x-1\right)\left(x+24\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-24. To find a and b, set up a system to be solved.

$$a+b=23 ab=1\left(-24\right)=-24$$

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 -24.

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

Calculate the sum for each pair.

$$-1+24=23 -2+12=10 -3+8=5 -4+6=2$$

The solution is the pair that gives sum 23.

$$a=-1 b=24$$

Rewrite x^{2}+23x-24 as \left(x^{2}-x\right)+\left(24x-24\right).

$$\left(x^{2}-x\right)+\left(24x-24\right)$$

Factor out x in the first and 24 in the second group.

$$x\left(x-1\right)+24\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(x+24\right)$$

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.

$$x^{2}+23x-24=0$$

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.

$$x=\frac{-23±\sqrt{23^{2}-4\left(-24\right)}}{2}$$

Square 23.

$$x=\frac{-23±\sqrt{529-4\left(-24\right)}}{2}$$

Multiply -4 times -24.

$$x=\frac{-23±\sqrt{529+96}}{2}$$

Add 529 to 96.

$$x=\frac{-23±\sqrt{625}}{2}$$

Take the square root of 625.

$$x=\frac{-23±25}{2}$$

Now solve the equation x=\frac{-23±25}{2} when ± is plus. Add -23 to 25.

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

Divide 2 by 2.

$$x=1$$

Now solve the equation x=\frac{-23±25}{2} when ± is minus. Subtract 25 from -23.

$$x=-\frac{48}{2}$$

Divide -48 by 2.

$$x=-24$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -24 for x_{2}.

$$x^{2}+23x-24=\left(x-1\right)\left(x-\left(-24\right)\right)$$

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

$$x^{2}+23x-24=\left(x-1\right)\left(x+24\right)$$

Evaluate

$$\left(x-1\right)\left(x+24\right)$$