Solve y^2+15y-16

$${ y }^{ 2 } +15y-16$$

Factor

$$\left(y-1\right)\left(y+16\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-16. To find a and b, set up a system to be solved.

$$a+b=15 ab=1\left(-16\right)=-16$$

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 -16.

$$-1,16 -2,8 -4,4$$

Calculate the sum for each pair.

$$-1+16=15 -2+8=6 -4+4=0$$

The solution is the pair that gives sum 15.

$$a=-1 b=16$$

Rewrite y^{2}+15y-16 as \left(y^{2}-y\right)+\left(16y-16\right).

$$\left(y^{2}-y\right)+\left(16y-16\right)$$

Factor out y in the first and 16 in the second group.

$$y\left(y-1\right)+16\left(y-1\right)$$

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

$$\left(y-1\right)\left(y+16\right)$$

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.

$$y^{2}+15y-16=0$$

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.

$$y=\frac{-15±\sqrt{15^{2}-4\left(-16\right)}}{2}$$

Square 15.

$$y=\frac{-15±\sqrt{225-4\left(-16\right)}}{2}$$

Multiply -4 times -16.

$$y=\frac{-15±\sqrt{225+64}}{2}$$

Add 225 to 64.

$$y=\frac{-15±\sqrt{289}}{2}$$

Take the square root of 289.

$$y=\frac{-15±17}{2}$$

Now solve the equation y=\frac{-15±17}{2} when ± is plus. Add -15 to 17.

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

Divide 2 by 2.

$$y=1$$

Now solve the equation y=\frac{-15±17}{2} when ± is minus. Subtract 17 from -15.

$$y=-\frac{32}{2}$$

Divide -32 by 2.

$$y=-16$$

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 -16 for x_{2}.

$$y^{2}+15y-16=\left(y-1\right)\left(y-\left(-16\right)\right)$$

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

$$y^{2}+15y-16=\left(y-1\right)\left(y+16\right)$$

Evaluate

$$\left(y-1\right)\left(y+16\right)$$