Solve 8+7y-y^2

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

Factor

$$-\left(y-8\right)\left(y+1\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

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

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.

$$a+b=7 ab=-8=-8$$

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 7.

$$a=8 b=-1$$

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

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

Factor out -y in the first and -1 in the second group.

$$-y\left(y-8\right)-\left(y-8\right)$$

Factor out common term y-8 by using distributive property.

$$\left(y-8\right)\left(-y-1\right)$$

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.

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

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.

$$y=\frac{-7±\sqrt{7^{2}-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Square 7.

$$y=\frac{-7±\sqrt{49-4\left(-1\right)\times 8}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-7±\sqrt{49+4\times 8}}{2\left(-1\right)}$$

Multiply 4 times 8.

$$y=\frac{-7±\sqrt{49+32}}{2\left(-1\right)}$$

Add 49 to 32.

$$y=\frac{-7±\sqrt{81}}{2\left(-1\right)}$$

Take the square root of 81.

$$y=\frac{-7±9}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{-7±9}{-2}$$

Now solve the equation y=\frac{-7±9}{-2} when ± is plus. Add -7 to 9.

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

Divide 2 by -2.

$$y=-1$$

Now solve the equation y=\frac{-7±9}{-2} when ± is minus. Subtract 9 from -7.

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

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 -1 for x_{1} and 8 for x_{2}.

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

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

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

Evaluate

$$-\left(y-8\right)\left(y+1\right)$$