Solve x^2-23x-24

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

Factor

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

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 negative, the negative number has greater absolute value than the positive. 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.

$$a=-24 b=1$$

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

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

Factor out x in x^{2}-24x.

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

Factor out common term x-24 by using distributive property.

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

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.

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

Square -23.

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

Multiply -4 times -24.

$$x=\frac{-\left(-23\right)±\sqrt{529+96}}{2}$$

Add 529 to 96.

$$x=\frac{-\left(-23\right)±\sqrt{625}}{2}$$

Take the square root of 625.

$$x=\frac{-\left(-23\right)±25}{2}$$

The opposite of -23 is 23.

$$x=\frac{23±25}{2}$$

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

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

Divide 48 by 2.

$$x=24$$

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

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

Divide -2 by 2.

$$x=-1$$

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

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

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

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

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 -23x -24 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = 23 rs = -24$$

Two numbers r and s sum up to 23 exactly when the average of the two numbers is \frac{1}{2}*23 = \frac{23}{2}. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{23}{2} - u s = \frac{23}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -24

$$(\frac{23}{2} - u) (\frac{23}{2} + u) = -24$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$\frac{529}{4} - u^2 = -24$$

Simplify the expression by subtracting \frac{529}{4} on both sides

$$-u^2 = -24-\frac{529}{4} = -\frac{625}{4}$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = \frac{625}{4} u = \pm\sqrt{\frac{625}{4}} = \pm \frac{25}{2}$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =\frac{23}{2} - \frac{25}{2} = -1 s = \frac{23}{2} + \frac{25}{2} = 24$$

Evaluate

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