Solve y^2+8y+15

$${ y }^{ 2 } +8y+15$$

Factor

$$\left(y+3\right)\left(y+5\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+15. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 15=15$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. List all such integer pairs that give product 15.

$$1,15 3,5$$

Calculate the sum for each pair.

$$1+15=16 3+5=8$$

The solution is the pair that gives sum 8.

$$a=3 b=5$$

Rewrite y^{2}+8y+15 as \left(y^{2}+3y\right)+\left(5y+15\right).

$$\left(y^{2}+3y\right)+\left(5y+15\right)$$

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

$$y\left(y+3\right)+5\left(y+3\right)$$

Factor out common term y+3 by using distributive property.

$$\left(y+3\right)\left(y+5\right)$$

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.

$$y^{2}+8y+15=0$$

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.

$$y=\frac{-8±\sqrt{8^{2}-4\times 15}}{2}$$

Square 8.

$$y=\frac{-8±\sqrt{64-4\times 15}}{2}$$

Multiply -4 times 15.

$$y=\frac{-8±\sqrt{64-60}}{2}$$

Add 64 to -60.

$$y=\frac{-8±\sqrt{4}}{2}$$

Take the square root of 4.

$$y=\frac{-8±2}{2}$$

Now solve the equation y=\frac{-8±2}{2} when ± is plus. Add -8 to 2.

$$y=-\frac{6}{2}$$

Divide -6 by 2.

$$y=-3$$

Now solve the equation y=\frac{-8±2}{2} when ± is minus. Subtract 2 from -8.

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

Divide -10 by 2.

$$y=-5$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and -5 for x_{2}.

$$y^{2}+8y+15=\left(y-\left(-3\right)\right)\left(y-\left(-5\right)\right)$$

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

$$y^{2}+8y+15=\left(y+3\right)\left(y+5\right)$$

Evaluate

$$\left(y+3\right)\left(y+5\right)$$