Solve m^2-2m-528=

$$m ^ { 2 } - 2 m - 528 =$$

Factor

$$\left(m-24\right)\left(m+22\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as m^{2}+am+bm-528. To find a and b, set up a system to be solved.

$$a+b=-2 ab=1\left(-528\right)=-528$$

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 -528.

$$1,-528 2,-264 3,-176 4,-132 6,-88 8,-66 11,-48 12,-44 16,-33 22,-24$$

Calculate the sum for each pair.

$$1-528=-527 2-264=-262 3-176=-173 4-132=-128 6-88=-82 8-66=-58 11-48=-37 12-44=-32 16-33=-17 22-24=-2$$

The solution is the pair that gives sum -2.

$$a=-24 b=22$$

Rewrite m^{2}-2m-528 as \left(m^{2}-24m\right)+\left(22m-528\right).

$$\left(m^{2}-24m\right)+\left(22m-528\right)$$

Factor out m in the first and 22 in the second group.

$$m\left(m-24\right)+22\left(m-24\right)$$

Factor out common term m-24 by using distributive property.

$$\left(m-24\right)\left(m+22\right)$$

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.

$$m^{2}-2m-528=0$$

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.

$$m=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-528\right)}}{2}$$

Square -2.

$$m=\frac{-\left(-2\right)±\sqrt{4-4\left(-528\right)}}{2}$$

Multiply -4 times -528.

$$m=\frac{-\left(-2\right)±\sqrt{4+2112}}{2}$$

Add 4 to 2112.

$$m=\frac{-\left(-2\right)±\sqrt{2116}}{2}$$

Take the square root of 2116.

$$m=\frac{-\left(-2\right)±46}{2}$$

The opposite of -2 is 2.

$$m=\frac{2±46}{2}$$

Now solve the equation m=\frac{2±46}{2} when ± is plus. Add 2 to 46.

$$m=\frac{48}{2}$$

Divide 48 by 2.

$$m=24$$

Now solve the equation m=\frac{2±46}{2} when ± is minus. Subtract 46 from 2.

$$m=-\frac{44}{2}$$

Divide -44 by 2.

$$m=-22$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 24 for x_{1} and -22 for x_{2}.

$$m^{2}-2m-528=\left(m-24\right)\left(m-\left(-22\right)\right)$$

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

$$m^{2}-2m-528=\left(m-24\right)\left(m+22\right)$$

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.

$$x ^ 2 -2x -528 = 0$$

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

$$r + s = 2 rs = -528$$

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 = -528

$$(1 - u) (1 + u) = -528$$

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

$$1 - u^2 = -528$$

Simplify the expression by subtracting 1 on both sides

$$-u^2 = -528-1 = -529$$

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

$$u^2 = 529 u = \pm\sqrt{529} = \pm 23$$

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

$$r =1 - 23 = -22 s = 1 + 23 = 24$$

Evaluate

$$\left(m-24\right)\left(m+22\right)$$