Solve y^2+8y+7

$${ y }^{ 2 } +8y+7$$

Factor

$$\left(y+1\right)\left(y+7\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by+7. To find a and b, set up a system to be solved.

$$a+b=8 ab=1\times 7=7$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. The only such pair is the system solution.

$$a=1 b=7$$

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

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

Factor out y in the first and 7 in the second group.

$$y\left(y+1\right)+7\left(y+1\right)$$

Factor out common term y+1 by using distributive property.

$$\left(y+1\right)\left(y+7\right)$$

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.

$$y^{2}+8y+7=0$$

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.

$$y=\frac{-8±\sqrt{8^{2}-4\times 7}}{2}$$

Square 8.

$$y=\frac{-8±\sqrt{64-4\times 7}}{2}$$

Multiply -4 times 7.

$$y=\frac{-8±\sqrt{64-28}}{2}$$

Add 64 to -28.

$$y=\frac{-8±\sqrt{36}}{2}$$

Take the square root of 36.

$$y=\frac{-8±6}{2}$$

Now solve the equation y=\frac{-8±6}{2} when ± is plus. Add -8 to 6.

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

Divide -2 by 2.

$$y=-1$$

Now solve the equation y=\frac{-8±6}{2} when ± is minus. Subtract 6 from -8.

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

Divide -14 by 2.

$$y=-7$$

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

$$y^{2}+8y+7=\left(y-\left(-1\right)\right)\left(y-\left(-7\right)\right)$$

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

$$y^{2}+8y+7=\left(y+1\right)\left(y+7\right)$$

Evaluate

$$\left(y+1\right)\left(y+7\right)$$