Solve x^2-32x-420

$$x ^ { 2 } - 32 x - 420$$

Factor

$$\left(x-42\right)\left(x+10\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-420. To find a and b, set up a system to be solved.

$$a+b=-32 ab=1\left(-420\right)=-420$$

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 -420.

$$1,-420 2,-210 3,-140 4,-105 5,-84 6,-70 7,-60 10,-42 12,-35 14,-30 15,-28 20,-21$$

Calculate the sum for each pair.

$$1-420=-419 2-210=-208 3-140=-137 4-105=-101 5-84=-79 6-70=-64 7-60=-53 10-42=-32 12-35=-23 14-30=-16 15-28=-13 20-21=-1$$

The solution is the pair that gives sum -32.

$$a=-42 b=10$$

Rewrite x^{2}-32x-420 as \left(x^{2}-42x\right)+\left(10x-420\right).

$$\left(x^{2}-42x\right)+\left(10x-420\right)$$

Factor out x in the first and 10 in the second group.

$$x\left(x-42\right)+10\left(x-42\right)$$

Factor out common term x-42 by using distributive property.

$$\left(x-42\right)\left(x+10\right)$$

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.

$$x^{2}-32x-420=0$$

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.

$$x=\frac{-\left(-32\right)±\sqrt{\left(-32\right)^{2}-4\left(-420\right)}}{2}$$

Square -32.

$$x=\frac{-\left(-32\right)±\sqrt{1024-4\left(-420\right)}}{2}$$

Multiply -4 times -420.

$$x=\frac{-\left(-32\right)±\sqrt{1024+1680}}{2}$$

Add 1024 to 1680.

$$x=\frac{-\left(-32\right)±\sqrt{2704}}{2}$$

Take the square root of 2704.

$$x=\frac{-\left(-32\right)±52}{2}$$

The opposite of -32 is 32.

$$x=\frac{32±52}{2}$$

Now solve the equation x=\frac{32±52}{2} when ± is plus. Add 32 to 52.

$$x=\frac{84}{2}$$

Divide 84 by 2.

$$x=42$$

Now solve the equation x=\frac{32±52}{2} when ± is minus. Subtract 52 from 32.

$$x=-\frac{20}{2}$$

Divide -20 by 2.

$$x=-10$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 42 for x_{1} and -10 for x_{2}.

$$x^{2}-32x-420=\left(x-42\right)\left(x-\left(-10\right)\right)$$

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

$$x^{2}-32x-420=\left(x-42\right)\left(x+10\right)$$

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.

$$x ^ 2 -32x -420 = 0$$

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

$$r + s = 32 rs = -420$$

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 = -420

$$(16 - u) (16 + u) = -420$$

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

$$256 - u^2 = -420$$

Simplify the expression by subtracting 256 on both sides

$$-u^2 = -420-256 = -676$$

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.

$$r =16 - 26 = -10 s = 16 + 26 = 42$$

Evaluate

$$\left(x-42\right)\left(x+10\right)$$