Solve y^2-8y-84

$${ y }^{ 2 } -8y-84$$

Factor

$$\left(y-14\right)\left(y+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-84. To find a and b, set up a system to be solved.

$$a+b=-8 ab=1\left(-84\right)=-84$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -84.

$$1,-84 2,-42 3,-28 4,-21 6,-14 7,-12$$

Calculate the sum for each pair.

$$1-84=-83 2-42=-40 3-28=-25 4-21=-17 6-14=-8 7-12=-5$$

The solution is the pair that gives sum -8.

$$a=-14 b=6$$

Rewrite y^{2}-8y-84 as \left(y^{2}-14y\right)+\left(6y-84\right).

$$\left(y^{2}-14y\right)+\left(6y-84\right)$$

Factor out y in the first and 6 in the second group.

$$y\left(y-14\right)+6\left(y-14\right)$$

Factor out common term y-14 by using distributive property.

$$\left(y-14\right)\left(y+6\right)$$

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.

$$y^{2}-8y-84=0$$

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.

$$y=\frac{-\left(-8\right)±\sqrt{\left(-8\right)^{2}-4\left(-84\right)}}{2}$$

Square -8.

$$y=\frac{-\left(-8\right)±\sqrt{64-4\left(-84\right)}}{2}$$

Multiply -4 times -84.

$$y=\frac{-\left(-8\right)±\sqrt{64+336}}{2}$$

Add 64 to 336.

$$y=\frac{-\left(-8\right)±\sqrt{400}}{2}$$

Take the square root of 400.

$$y=\frac{-\left(-8\right)±20}{2}$$

The opposite of -8 is 8.

$$y=\frac{8±20}{2}$$

Now solve the equation y=\frac{8±20}{2} when ± is plus. Add 8 to 20.

$$y=\frac{28}{2}$$

Divide 28 by 2.

$$y=14$$

Now solve the equation y=\frac{8±20}{2} when ± is minus. Subtract 20 from 8.

$$y=-\frac{12}{2}$$

Divide -12 by 2.

$$y=-6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 14 for x_{1} and -6 for x_{2}.

$$y^{2}-8y-84=\left(y-14\right)\left(y-\left(-6\right)\right)$$

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

$$y^{2}-8y-84=\left(y-14\right)\left(y+6\right)$$

Evaluate

$$\left(y-14\right)\left(y+6\right)$$