Solve x^2-32x-2448

$$x ^ { 2 } - 32 x - 2448$$

Factor

$$\left(x-68\right)\left(x+36\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-2448. To find a and b, set up a system to be solved.

$$a+b=-32 ab=1\left(-2448\right)=-2448$$

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 -2448.

$$1,-2448 2,-1224 3,-816 4,-612 6,-408 8,-306 9,-272 12,-204 16,-153 17,-144 18,-136 24,-102 34,-72 36,-68 48,-51$$

Calculate the sum for each pair.

$$1-2448=-2447 2-1224=-1222 3-816=-813 4-612=-608 6-408=-402 8-306=-298 9-272=-263 12-204=-192 16-153=-137 17-144=-127 18-136=-118 24-102=-78 34-72=-38 36-68=-32 48-51=-3$$

The solution is the pair that gives sum -32.

$$a=-68 b=36$$

Rewrite x^{2}-32x-2448 as \left(x^{2}-68x\right)+\left(36x-2448\right).

$$\left(x^{2}-68x\right)+\left(36x-2448\right)$$

Factor out x in the first and 36 in the second group.

$$x\left(x-68\right)+36\left(x-68\right)$$

Factor out common term x-68 by using distributive property.

$$\left(x-68\right)\left(x+36\right)$$

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.

$$x^{2}-32x-2448=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-2448\right)}}{2}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\left(-2448\right)}}{2}$$

Multiply -4 times -2448.

$$x=\frac{-\left(-32\right)±\sqrt{1024+9792}}{2}$$

Add 1024 to 9792.

$$x=\frac{-\left(-32\right)±\sqrt{10816}}{2}$$

Take the square root of 10816.

$$x=\frac{-\left(-32\right)±104}{2}$$

The opposite of -32 is 32.

$$x=\frac{32±104}{2}$$

Now solve the equation x=\frac{32±104}{2} when ± is plus. Add 32 to 104.

$$x=\frac{136}{2}$$

Divide 136 by 2.

$$x=68$$

Now solve the equation x=\frac{32±104}{2} when ± is minus. Subtract 104 from 32.

$$x=-\frac{72}{2}$$

Divide -72 by 2.

$$x=-36$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 68 for x_{1} and -36 for x_{2}.

$$x^{2}-32x-2448=\left(x-68\right)\left(x-\left(-36\right)\right)$$

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

$$x^{2}-32x-2448=\left(x-68\right)\left(x+36\right)$$

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.

$$x ^ 2 -32x -2448 = 0$$

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

$$r + s = 32 rs = -2448$$

Two numbers r and s sum up to 32 exactly when the average of the two numbers is \frac{1}{2}*32 = 16. 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>

$$r = 16 - u s = 16 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -2448

$$(16 - u) (16 + u) = -2448$$

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

$$256 - u^2 = -2448$$

Simplify the expression by subtracting 256 on both sides

$$-u^2 = -2448-256 = -2704$$

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

$$u^2 = 2704 u = \pm\sqrt{2704} = \pm 52$$

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

$$r =16 - 52 = -36 s = 16 + 52 = 68$$

Evaluate

$$\left(x-68\right)\left(x+36\right)$$