Solve (4y^2+4y-24)

$$( 4 y ^ { 2 } + 4 y - 24 )$$

Factor

$$4\left(y-2\right)\left(y+3\right)$$

Solution Steps

Factor out 4.

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

Consider y^{2}+y-6. Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-6. To find a and b, set up a system to be solved.

$$a+b=1 ab=1\left(-6\right)=-6$$

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 -6.

$$-1,6 -2,3$$

Calculate the sum for each pair.

$$-1+6=5 -2+3=1$$

The solution is the pair that gives sum 1.

$$a=-2 b=3$$

Rewrite y^{2}+y-6 as \left(y^{2}-2y\right)+\left(3y-6\right).

$$\left(y^{2}-2y\right)+\left(3y-6\right)$$

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

$$y\left(y-2\right)+3\left(y-2\right)$$

Factor out common term y-2 by using distributive property.

$$\left(y-2\right)\left(y+3\right)$$

Rewrite the complete factored expression.

$$4\left(y-2\right)\left(y+3\right)$$

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.

$$4y^{2}+4y-24=0$$

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.

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

Square 4.

$$y=\frac{-4±\sqrt{16-4\times 4\left(-24\right)}}{2\times 4}$$

Multiply -4 times 4.

$$y=\frac{-4±\sqrt{16-16\left(-24\right)}}{2\times 4}$$

Multiply -16 times -24.

$$y=\frac{-4±\sqrt{16+384}}{2\times 4}$$

Add 16 to 384.

$$y=\frac{-4±\sqrt{400}}{2\times 4}$$

Take the square root of 400.

$$y=\frac{-4±20}{2\times 4}$$

Multiply 2 times 4.

$$y=\frac{-4±20}{8}$$

Now solve the equation y=\frac{-4±20}{8} when ± is plus. Add -4 to 20.

$$y=\frac{16}{8}$$

Divide 16 by 8.

$$y=2$$

Now solve the equation y=\frac{-4±20}{8} when ± is minus. Subtract 20 from -4.

$$y=-\frac{24}{8}$$

Divide -24 by 8.

$$y=-3$$

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

$$4y^{2}+4y-24=4\left(y-2\right)\left(y-\left(-3\right)\right)$$

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

$$4y^{2}+4y-24=4\left(y-2\right)\left(y+3\right)$$

Evaluate

$$4\left(y-2\right)\left(y+3\right)$$