Solve a^2-124a-512=0

$$a ^ { 2 } - 124 a - 512 = 0$$

Solve for a

$$a=-4$$

$$a=128$$

Steps Using Factoring

To solve the equation, factor a^{2}-124a-512 using formula a^{2}+\left(a+b\right)a+ab=\left(a+a\right)\left(a+b\right). To find a and b, set up a system to be solved.

$$a+b=-124 ab=-512$$

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 -512.

$$1,-512 2,-256 4,-128 8,-64 16,-32$$

Calculate the sum for each pair.

$$1-512=-511 2-256=-254 4-128=-124 8-64=-56 16-32=-16$$

The solution is the pair that gives sum -124.

$$a=-128 b=4$$

Rewrite factored expression \left(a+a\right)\left(a+b\right) using the obtained values.

$$\left(a-128\right)\left(a+4\right)$$

To find equation solutions, solve a-128=0 and a+4=0.

$$a=128 a=-4$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as a^{2}+aa+ba-512. To find a and b, set up a system to be solved.

$$a+b=-124 ab=1\left(-512\right)=-512$$

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 -512.

$$1,-512 2,-256 4,-128 8,-64 16,-32$$

Calculate the sum for each pair.

$$1-512=-511 2-256=-254 4-128=-124 8-64=-56 16-32=-16$$

The solution is the pair that gives sum -124.

$$a=-128 b=4$$

Rewrite a^{2}-124a-512 as \left(a^{2}-128a\right)+\left(4a-512\right).

$$\left(a^{2}-128a\right)+\left(4a-512\right)$$

Factor out a in the first and 4 in the second group.

$$a\left(a-128\right)+4\left(a-128\right)$$

Factor out common term a-128 by using distributive property.

$$\left(a-128\right)\left(a+4\right)$$

To find equation solutions, solve a-128=0 and a+4=0.

$$a=128 a=-4$$

Steps Using the Quadratic Formula

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.

$$a^{2}-124a-512=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -124 for b, and -512 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$a=\frac{-\left(-124\right)±\sqrt{\left(-124\right)^{2}-4\left(-512\right)}}{2}$$

Square -124.

$$a=\frac{-\left(-124\right)±\sqrt{15376-4\left(-512\right)}}{2}$$

Multiply -4 times -512.

$$a=\frac{-\left(-124\right)±\sqrt{15376+2048}}{2}$$

Add 15376 to 2048.

$$a=\frac{-\left(-124\right)±\sqrt{17424}}{2}$$

Take the square root of 17424.

$$a=\frac{-\left(-124\right)±132}{2}$$

The opposite of -124 is 124.

$$a=\frac{124±132}{2}$$

Now solve the equation a=\frac{124±132}{2} when ± is plus. Add 124 to 132.

$$a=\frac{256}{2}$$

Divide 256 by 2.

$$a=128$$

Now solve the equation a=\frac{124±132}{2} when ± is minus. Subtract 132 from 124.

$$a=-\frac{8}{2}$$

Divide -8 by 2.

$$a=-4$$

The equation is now solved.

$$a=128 a=-4$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$a^{2}-124a-512=0$$

Add 512 to both sides of the equation.

$$a^{2}-124a-512-\left(-512\right)=-\left(-512\right)$$

Subtracting -512 from itself leaves 0.

$$a^{2}-124a=-\left(-512\right)$$

Subtract -512 from 0.

$$a^{2}-124a=512$$

Divide -124, the coefficient of the x term, by 2 to get -62. Then add the square of -62 to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$a^{2}-124a+\left(-62\right)^{2}=512+\left(-62\right)^{2}$$

Square -62.

$$a^{2}-124a+3844=512+3844$$

Add 512 to 3844.

$$a^{2}-124a+3844=4356$$

Factor a^{2}-124a+3844. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(a-62\right)^{2}=4356$$

Take the square root of both sides of the equation.

$$\sqrt{\left(a-62\right)^{2}}=\sqrt{4356}$$

Simplify.

$$a-62=66 a-62=-66$$

Add 62 to both sides of the equation.

$$a=128 a=-4$$

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.

$$x ^ 2 -124x -512 = 0$$

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

$$r + s = 124 rs = -512$$

Two numbers r and s sum up to 124 exactly when the average of the two numbers is \frac{1}{2}*124 = 62. 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>

$$r = 62 - u s = 62 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -512

$$(62 - u) (62 + u) = -512$$

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

$$3844 - u^2 = -512$$

Simplify the expression by subtracting 3844 on both sides

$$-u^2 = -512-3844 = -4356$$

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

$$u^2 = 4356 u = \pm\sqrt{4356} = \pm 66$$

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

$$r =62 - 66 = -4 s = 62 + 66 = 128$$