Solve (y^2+15y+56)

$$( y ^ { 2 } + 15 y + 56 )$$

Factor

$$\left(y+7\right)\left(y+8\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+56. To find a and b, set up a system to be solved.

$$a+b=15 ab=1\times 56=56$$

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 56.

$$1,56 2,28 4,14 7,8$$

Calculate the sum for each pair.

$$1+56=57 2+28=30 4+14=18 7+8=15$$

The solution is the pair that gives sum 15.

$$a=7 b=8$$

Rewrite y^{2}+15y+56 as \left(y^{2}+7y\right)+\left(8y+56\right).

$$\left(y^{2}+7y\right)+\left(8y+56\right)$$

Factor out y in the first and 8 in the second group.

$$y\left(y+7\right)+8\left(y+7\right)$$

Factor out common term y+7 by using distributive property.

$$\left(y+7\right)\left(y+8\right)$$

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.

$$y^{2}+15y+56=0$$

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.

$$y=\frac{-15±\sqrt{15^{2}-4\times 56}}{2}$$

Square 15.

$$y=\frac{-15±\sqrt{225-4\times 56}}{2}$$

Multiply -4 times 56.

$$y=\frac{-15±\sqrt{225-224}}{2}$$

Add 225 to -224.

$$y=\frac{-15±\sqrt{1}}{2}$$

Take the square root of 1.

$$y=\frac{-15±1}{2}$$

Now solve the equation y=\frac{-15±1}{2} when ± is plus. Add -15 to 1.

$$y=-\frac{14}{2}$$

Divide -14 by 2.

$$y=-7$$

Now solve the equation y=\frac{-15±1}{2} when ± is minus. Subtract 1 from -15.

$$y=-\frac{16}{2}$$

Divide -16 by 2.

$$y=-8$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -7 for x_{1} and -8 for x_{2}.

$$y^{2}+15y+56=\left(y-\left(-7\right)\right)\left(y-\left(-8\right)\right)$$

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

$$y^{2}+15y+56=\left(y+7\right)\left(y+8\right)$$

Evaluate

$$\left(y+7\right)\left(y+8\right)$$