Solve x^2-20x+84

$$x ^ { 2 } - 20 x + 84$$

Factor

$$\left(x-14\right)\left(x-6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+84. To find a and b, set up a system to be solved.

$$a+b=-20 ab=1\times 84=84$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. List all such integer pairs that give product 84.

$$-1,-84 -2,-42 -3,-28 -4,-21 -6,-14 -7,-12$$

Calculate the sum for each pair.

$$-1-84=-85 -2-42=-44 -3-28=-31 -4-21=-25 -6-14=-20 -7-12=-19$$

The solution is the pair that gives sum -20.

$$a=-14 b=-6$$

Rewrite x^{2}-20x+84 as \left(x^{2}-14x\right)+\left(-6x+84\right).

$$\left(x^{2}-14x\right)+\left(-6x+84\right)$$

Factor out x in the first and -6 in the second group.

$$x\left(x-14\right)-6\left(x-14\right)$$

Factor out common term x-14 by using distributive property.

$$\left(x-14\right)\left(x-6\right)$$

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.

$$x^{2}-20x+84=0$$

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.

$$x=\frac{-\left(-20\right)±\sqrt{\left(-20\right)^{2}-4\times 84}}{2}$$

Square -20.

$$x=\frac{-\left(-20\right)±\sqrt{400-4\times 84}}{2}$$

Multiply -4 times 84.

$$x=\frac{-\left(-20\right)±\sqrt{400-336}}{2}$$

Add 400 to -336.

$$x=\frac{-\left(-20\right)±\sqrt{64}}{2}$$

Take the square root of 64.

$$x=\frac{-\left(-20\right)±8}{2}$$

The opposite of -20 is 20.

$$x=\frac{20±8}{2}$$

Now solve the equation x=\frac{20±8}{2} when ± is plus. Add 20 to 8.

$$x=\frac{28}{2}$$

Divide 28 by 2.

$$x=14$$

Now solve the equation x=\frac{20±8}{2} when ± is minus. Subtract 8 from 20.

$$x=\frac{12}{2}$$

Divide 12 by 2.

$$x=6$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 14 for x_{1} and 6 for x_{2}.

$$x^{2}-20x+84=\left(x-14\right)\left(x-6\right)$$

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.

$$x ^ 2 -20x +84 = 0$$

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

$$r + s = 20 rs = 84$$

Two numbers r and s sum up to 20 exactly when the average of the two numbers is \frac{1}{2}*20 = 10. 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>

$$r = 10 - u s = 10 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 84

$$(10 - u) (10 + u) = 84$$

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

$$100 - u^2 = 84$$

Simplify the expression by subtracting 100 on both sides

$$-u^2 = 84-100 = -16$$

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

$$u^2 = 16 u = \pm\sqrt{16} = \pm 4$$

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

$$r =10 - 4 = 6 s = 10 + 4 = 14$$

Evaluate

$$\left(x-14\right)\left(x-6\right)$$