Solve x^2-28x-480

$$x ^ { 2 } - 28 x - 480$$

Factor

$$\left(x-40\right)\left(x+12\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-480. To find a and b, set up a system to be solved.

$$a+b=-28 ab=1\left(-480\right)=-480$$

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 -480.

$$1,-480 2,-240 3,-160 4,-120 5,-96 6,-80 8,-60 10,-48 12,-40 15,-32 16,-30 20,-24$$

Calculate the sum for each pair.

$$1-480=-479 2-240=-238 3-160=-157 4-120=-116 5-96=-91 6-80=-74 8-60=-52 10-48=-38 12-40=-28 15-32=-17 16-30=-14 20-24=-4$$

The solution is the pair that gives sum -28.

$$a=-40 b=12$$

Rewrite x^{2}-28x-480 as \left(x^{2}-40x\right)+\left(12x-480\right).

$$\left(x^{2}-40x\right)+\left(12x-480\right)$$

Factor out x in the first and 12 in the second group.

$$x\left(x-40\right)+12\left(x-40\right)$$

Factor out common term x-40 by using distributive property.

$$\left(x-40\right)\left(x+12\right)$$

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.

$$x^{2}-28x-480=0$$

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.

$$x=\frac{-\left(-28\right)±\sqrt{\left(-28\right)^{2}-4\left(-480\right)}}{2}$$

Square -28.

$$x=\frac{-\left(-28\right)±\sqrt{784-4\left(-480\right)}}{2}$$

Multiply -4 times -480.

$$x=\frac{-\left(-28\right)±\sqrt{784+1920}}{2}$$

Add 784 to 1920.

$$x=\frac{-\left(-28\right)±\sqrt{2704}}{2}$$

Take the square root of 2704.

$$x=\frac{-\left(-28\right)±52}{2}$$

The opposite of -28 is 28.

$$x=\frac{28±52}{2}$$

Now solve the equation x=\frac{28±52}{2} when ± is plus. Add 28 to 52.

$$x=\frac{80}{2}$$

Divide 80 by 2.

$$x=40$$

Now solve the equation x=\frac{28±52}{2} when ± is minus. Subtract 52 from 28.

$$x=-\frac{24}{2}$$

Divide -24 by 2.

$$x=-12$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 40 for x_{1} and -12 for x_{2}.

$$x^{2}-28x-480=\left(x-40\right)\left(x-\left(-12\right)\right)$$

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

$$x^{2}-28x-480=\left(x-40\right)\left(x+12\right)$$

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.

$$x ^ 2 -28x -480 = 0$$

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

$$r + s = 28 rs = -480$$

Two numbers r and s sum up to 28 exactly when the average of the two numbers is \frac{1}{2}*28 = 14. 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>

$$r = 14 - u s = 14 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -480

$$(14 - u) (14 + u) = -480$$

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

$$196 - u^2 = -480$$

Simplify the expression by subtracting 196 on both sides

$$-u^2 = -480-196 = -676$$

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

$$u^2 = 676 u = \pm\sqrt{676} = \pm 26$$

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

$$r =14 - 26 = -12 s = 14 + 26 = 40$$

Evaluate

$$\left(x-40\right)\left(x+12\right)$$