Solve -2y^2+2y+112

$$-2 { y }^{ 2 } +2y+112$$

Factor

$$-2\left(y-8\right)\left(y+7\right)$$

Solution Steps

Factor out 2.

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

Consider -y^{2}+y+56. 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.

$$a+b=1 ab=-56=-56$$

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 -56.

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

Calculate the sum for each pair.

$$-1+56=55 -2+28=26 -4+14=10 -7+8=1$$

The solution is the pair that gives sum 1.

$$a=8 b=-7$$

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

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

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

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

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

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

Rewrite the complete factored expression.

$$2\left(y-8\right)\left(-y-7\right)$$

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.

$$-2y^{2}+2y+112=0$$

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.

$$y=\frac{-2±\sqrt{2^{2}-4\left(-2\right)\times 112}}{2\left(-2\right)}$$

Square 2.

$$y=\frac{-2±\sqrt{4-4\left(-2\right)\times 112}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$y=\frac{-2±\sqrt{4+8\times 112}}{2\left(-2\right)}$$

Multiply 8 times 112.

$$y=\frac{-2±\sqrt{4+896}}{2\left(-2\right)}$$

Add 4 to 896.

$$y=\frac{-2±\sqrt{900}}{2\left(-2\right)}$$

Take the square root of 900.

$$y=\frac{-2±30}{2\left(-2\right)}$$

Multiply 2 times -2.

$$y=\frac{-2±30}{-4}$$

Now solve the equation y=\frac{-2±30}{-4} when ± is plus. Add -2 to 30.

$$y=\frac{28}{-4}$$

Divide 28 by -4.

$$y=-7$$

Now solve the equation y=\frac{-2±30}{-4} when ± is minus. Subtract 30 from -2.

$$y=-\frac{32}{-4}$$

Divide -32 by -4.

$$y=8$$

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}.

$$-2y^{2}+2y+112=-2\left(y-\left(-7\right)\right)\left(y-8\right)$$

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

$$-2y^{2}+2y+112=-2\left(y+7\right)\left(y-8\right)$$

Evaluate

$$-2\left(y-8\right)\left(y+7\right)$$