Solve x^2-26x+168=

$$x ^ { 2 } - 26 x + 168 =$$

Factor

$$\left(x-14\right)\left(x-12\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+168. To find a and b, set up a system to be solved.

$$a+b=-26 ab=1\times 168=168$$

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 168.

$$-1,-168 -2,-84 -3,-56 -4,-42 -6,-28 -7,-24 -8,-21 -12,-14$$

Calculate the sum for each pair.

$$-1-168=-169 -2-84=-86 -3-56=-59 -4-42=-46 -6-28=-34 -7-24=-31 -8-21=-29 -12-14=-26$$

The solution is the pair that gives sum -26.

$$a=-14 b=-12$$

Rewrite x^{2}-26x+168 as \left(x^{2}-14x\right)+\left(-12x+168\right).

$$\left(x^{2}-14x\right)+\left(-12x+168\right)$$

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

$$x\left(x-14\right)-12\left(x-14\right)$$

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

$$\left(x-14\right)\left(x-12\right)$$

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.

$$x^{2}-26x+168=0$$

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.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 168}}{2}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\times 168}}{2}$$

Multiply -4 times 168.

$$x=\frac{-\left(-26\right)±\sqrt{676-672}}{2}$$

Add 676 to -672.

$$x=\frac{-\left(-26\right)±\sqrt{4}}{2}$$

Take the square root of 4.

$$x=\frac{-\left(-26\right)±2}{2}$$

The opposite of -26 is 26.

$$x=\frac{26±2}{2}$$

Now solve the equation x=\frac{26±2}{2} when ± is plus. Add 26 to 2.

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

Divide 28 by 2.

$$x=14$$

Now solve the equation x=\frac{26±2}{2} when ± is minus. Subtract 2 from 26.

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

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 14 for x_{1} and 12 for x_{2}.

$$x^{2}-26x+168=\left(x-14\right)\left(x-12\right)$$

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.

$$x ^ 2 -26x +168 = 0$$

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

$$r + s = 26 rs = 168$$

Two numbers r and s sum up to 26 exactly when the average of the two numbers is \frac{1}{2}*26 = 13. 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>

$$r = 13 - u s = 13 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 168

$$(13 - u) (13 + u) = 168$$

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

$$169 - u^2 = 168$$

Simplify the expression by subtracting 169 on both sides

$$-u^2 = 168-169 = -1$$

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

$$u^2 = 1 u = \pm\sqrt{1} = \pm 1$$

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

$$r =13 - 1 = 12 s = 13 + 1 = 14$$

Evaluate

$$\left(x-14\right)\left(x-12\right)$$