Solve y^2-2y-24

$${ y }^{ 2 } -2y-24$$

Factor

$$\left(y-6\right)\left(y+4\right)$$

Solution Steps

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

$$a+b=-2 ab=1\left(-24\right)=-24$$

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 -2.

$$a=-6 b=4$$

Rewrite y^{2}-2y-24 as \left(y^{2}-6y\right)+\left(4y-24\right).

$$\left(y^{2}-6y\right)+\left(4y-24\right)$$

Factor out y in the first and 4 in the second group.

$$y\left(y-6\right)+4\left(y-6\right)$$

Factor out common term y-6 by using distributive property.

$$\left(y-6\right)\left(y+4\right)$$

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.

$$y^{2}-2y-24=0$$

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.

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

Square -2.

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

Multiply -4 times -24.

$$y=\frac{-\left(-2\right)±\sqrt{4+96}}{2}$$

Add 4 to 96.

$$y=\frac{-\left(-2\right)±\sqrt{100}}{2}$$

Take the square root of 100.

$$y=\frac{-\left(-2\right)±10}{2}$$

The opposite of -2 is 2.

$$y=\frac{2±10}{2}$$

Now solve the equation y=\frac{2±10}{2} when ± is plus. Add 2 to 10.

$$y=\frac{12}{2}$$

Divide 12 by 2.

$$y=6$$

Now solve the equation y=\frac{2±10}{2} when ± is minus. Subtract 10 from 2.

$$y=-\frac{8}{2}$$

Divide -8 by 2.

$$y=-4$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and -4 for x_{2}.

$$y^{2}-2y-24=\left(y-6\right)\left(y-\left(-4\right)\right)$$

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

$$y^{2}-2y-24=\left(y-6\right)\left(y+4\right)$$

Evaluate

$$\left(y-6\right)\left(y+4\right)$$