Solve y^2+2y-360

$$y ^ { 2 } + 2 y - 360$$

Factor

$$\left(y-18\right)\left(y+20\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-360. To find a and b, set up a system to be solved.

$$a+b=2 ab=1\left(-360\right)=-360$$

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 -360.

$$-1,360 -2,180 -3,120 -4,90 -5,72 -6,60 -8,45 -9,40 -10,36 -12,30 -15,24 -18,20$$

Calculate the sum for each pair.

$$-1+360=359 -2+180=178 -3+120=117 -4+90=86 -5+72=67 -6+60=54 -8+45=37 -9+40=31 -10+36=26 -12+30=18 -15+24=9 -18+20=2$$

The solution is the pair that gives sum 2.

$$a=-18 b=20$$

Rewrite y^{2}+2y-360 as \left(y^{2}-18y\right)+\left(20y-360\right).

$$\left(y^{2}-18y\right)+\left(20y-360\right)$$

Factor out y in the first and 20 in the second group.

$$y\left(y-18\right)+20\left(y-18\right)$$

Factor out common term y-18 by using distributive property.

$$\left(y-18\right)\left(y+20\right)$$

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.

$$y^{2}+2y-360=0$$

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.

$$y=\frac{-2±\sqrt{2^{2}-4\left(-360\right)}}{2}$$

Square 2.

$$y=\frac{-2±\sqrt{4-4\left(-360\right)}}{2}$$

Multiply -4 times -360.

$$y=\frac{-2±\sqrt{4+1440}}{2}$$

Add 4 to 1440.

$$y=\frac{-2±\sqrt{1444}}{2}$$

Take the square root of 1444.

$$y=\frac{-2±38}{2}$$

Now solve the equation y=\frac{-2±38}{2} when ± is plus. Add -2 to 38.

$$y=\frac{36}{2}$$

Divide 36 by 2.

$$y=18$$

Now solve the equation y=\frac{-2±38}{2} when ± is minus. Subtract 38 from -2.

$$y=-\frac{40}{2}$$

Divide -40 by 2.

$$y=-20$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 18 for x_{1} and -20 for x_{2}.

$$y^{2}+2y-360=\left(y-18\right)\left(y-\left(-20\right)\right)$$

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

$$y^{2}+2y-360=\left(y-18\right)\left(y+20\right)$$

Steps Using Direct Factoring Method

Quadratic equations such as this one can be solved by a new direct factoring method that does not require guess work. To use the direct factoring method, the equation must be in the form x^2+Bx+C=0.

$$x ^ 2 +2x -360 = 0$$

Let r and s be the factors for the quadratic equation such that x^2+Bx+C=(x−r)(x−s) where sum of factors (r+s)=−B and the product of factors rs = C

$$r + s = -2 rs = -360$$

Two numbers r and s sum up to -2 exactly when the average of the two numbers is \frac{1}{2}*-2 = -1. You can also see that the midpoint of r and s corresponds to the axis of symmetry of the parabola represented by the quadratic equation y=x^2+Bx+C. The values of r and s are equidistant from the center by an unknown quantity u. Express r and s with respect to variable u. <div style='padding: 8px'><img src='https://opalmath.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = -1 - u s = -1 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -360

$$(-1 - u) (-1 + u) = -360$$

Simplify by expanding (a -b) (a + b) = a^2 – b^2

$$1 - u^2 = -360$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -360-1 = -361$$

Simplify the expression by multiplying -1 on both sides and take the square root to obtain the value of unknown variable u

$$u^2 = 361 u = \pm\sqrt{361} = \pm 19$$

The factors r and s are the solutions to the quadratic equation. Substitute the value of u to compute the r and s.

$$r =-1 - 19 = -20 s = -1 + 19 = 18$$

Evaluate

$$\left(y-18\right)\left(y+20\right)$$