Solve -y^2+2y+8

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

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -y^{2}+ay+by+8. To find a and b, set up a system to be solved.

$$a+b=2 ab=-8=-8$$

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 -8.

$$-1,8 -2,4$$

Calculate the sum for each pair.

$$-1+8=7 -2+4=2$$

The solution is the pair that gives sum 2.

$$a=4 b=-2$$

Rewrite -y^{2}+2y+8 as \left(-y^{2}+4y\right)+\left(-2y+8\right).

$$\left(-y^{2}+4y\right)+\left(-2y+8\right)$$

Factor out -y in the first and -2 in the second group.

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

Factor out common term y-4 by using distributive property.

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

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.

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

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.

$$y=\frac{-2±\sqrt{2^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square 2.

$$y=\frac{-2±\sqrt{4-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-2±\sqrt{4+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$y=\frac{-2±\sqrt{4+32}}{2\left(-1\right)}$$

Add 4 to 32.

$$y=\frac{-2±\sqrt{36}}{2\left(-1\right)}$$

Take the square root of 36.

$$y=\frac{-2±6}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-2±6}{-2}$$

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

$$y=\frac{4}{-2}$$

Divide 4 by -2.

$$y=-2$$

Now solve the equation y=\frac{-2±6}{-2} when ± is minus. Subtract 6 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 -2 for x_{1} and 4 for x_{2}.

$$-y^{2}+2y+8=-\left(y-\left(-2\right)\right)\left(y-4\right)$$

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

$$-y^{2}+2y+8=-\left(y+2\right)\left(y-4\right)$$

Evaluate

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