Solve y^2-48y-324

$$y ^ { 2 } - 48 y - 324$$

Factor

$$\left(y-54\right)\left(y+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as y^{2}+ay+by-324. To find a and b, set up a system to be solved.

$$a+b=-48 ab=1\left(-324\right)=-324$$

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 -324.

$$1,-324 2,-162 3,-108 4,-81 6,-54 9,-36 12,-27 18,-18$$

Calculate the sum for each pair.

$$1-324=-323 2-162=-160 3-108=-105 4-81=-77 6-54=-48 9-36=-27 12-27=-15 18-18=0$$

The solution is the pair that gives sum -48.

$$a=-54 b=6$$

Rewrite y^{2}-48y-324 as \left(y^{2}-54y\right)+\left(6y-324\right).

$$\left(y^{2}-54y\right)+\left(6y-324\right)$$

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

$$y\left(y-54\right)+6\left(y-54\right)$$

Factor out common term y-54 by using distributive property.

$$\left(y-54\right)\left(y+6\right)$$

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.

$$y^{2}-48y-324=0$$

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.

$$y=\frac{-\left(-48\right)±\sqrt{\left(-48\right)^{2}-4\left(-324\right)}}{2}$$

Square -48.

$$y=\frac{-\left(-48\right)±\sqrt{2304-4\left(-324\right)}}{2}$$

Multiply -4 times -324.

$$y=\frac{-\left(-48\right)±\sqrt{2304+1296}}{2}$$

Add 2304 to 1296.

$$y=\frac{-\left(-48\right)±\sqrt{3600}}{2}$$

Take the square root of 3600.

$$y=\frac{-\left(-48\right)±60}{2}$$

The opposite of -48 is 48.

$$y=\frac{48±60}{2}$$

Now solve the equation y=\frac{48±60}{2} when ± is plus. Add 48 to 60.

$$y=\frac{108}{2}$$

Divide 108 by 2.

$$y=54$$

Now solve the equation y=\frac{48±60}{2} when ± is minus. Subtract 60 from 48.

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

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 54 for x_{1} and -6 for x_{2}.

$$y^{2}-48y-324=\left(y-54\right)\left(y-\left(-6\right)\right)$$

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

$$y^{2}-48y-324=\left(y-54\right)\left(y+6\right)$$

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.

$$x ^ 2 -48x -324 = 0$$

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

$$r + s = 48 rs = -324$$

Two numbers r and s sum up to 48 exactly when the average of the two numbers is \frac{1}{2}*48 = 24. 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>

$$r = 24 - u s = 24 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -324

$$(24 - u) (24 + u) = -324$$

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

$$576 - u^2 = -324$$

Simplify the expression by subtracting 576 on both sides

$$-u^2 = -324-576 = -900$$

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

$$u^2 = 900 u = \pm\sqrt{900} = \pm 30$$

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

$$r =24 - 30 = -6 s = 24 + 30 = 54$$

Evaluate

$$\left(y-54\right)\left(y+6\right)$$