Solve y^2-64y-11520

$$y ^ { 2 } - 64 y - 11520$$

Factor

$$\left(y-144\right)\left(y+80\right)$$

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.

$$y^{2}-64y-11520=0$$

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.

$$y=\frac{-\left(-64\right)±\sqrt{\left(-64\right)^{2}-4\left(-11520\right)}}{2}$$

Square -64.

$$y=\frac{-\left(-64\right)±\sqrt{4096-4\left(-11520\right)}}{2}$$

Multiply -4 times -11520.

$$y=\frac{-\left(-64\right)±\sqrt{4096+46080}}{2}$$

Add 4096 to 46080.

$$y=\frac{-\left(-64\right)±\sqrt{50176}}{2}$$

Take the square root of 50176.

$$y=\frac{-\left(-64\right)±224}{2}$$

The opposite of -64 is 64.

$$y=\frac{64±224}{2}$$

Now solve the equation y=\frac{64±224}{2} when ± is plus. Add 64 to 224.

$$y=\frac{288}{2}$$

Divide 288 by 2.

$$y=144$$

Now solve the equation y=\frac{64±224}{2} when ± is minus. Subtract 224 from 64.

$$y=-\frac{160}{2}$$

Divide -160 by 2.

$$y=-80$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 144 for x_{1} and -80 for x_{2}.

$$y^{2}-64y-11520=\left(y-144\right)\left(y-\left(-80\right)\right)$$

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

$$y^{2}-64y-11520=\left(y-144\right)\left(y+80\right)$$

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.

$$x ^ 2 -64x -11520 = 0$$

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

$$r + s = 64 rs = -11520$$

Two numbers r and s sum up to 64 exactly when the average of the two numbers is \frac{1}{2}*64 = 32. 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>

$$r = 32 - u s = 32 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -11520

$$(32 - u) (32 + u) = -11520$$

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

$$1024 - u^2 = -11520$$

Simplify the expression by subtracting 1024 on both sides

$$-u^2 = -11520-1024 = -12544$$

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

$$u^2 = 12544 u = \pm\sqrt{12544} = \pm 112$$

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

$$r =32 - 112 = -80 s = 32 + 112 = 144$$

Evaluate

$$\left(y-144\right)\left(y+80\right)$$