Solve x^2-26x+48

$$x ^ { 2 } - 26 x + 48$$

Factor

$$\left(x-24\right)\left(x-2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+48. To find a and b, set up a system to be solved.

$$a+b=-26 ab=1\times 48=48$$

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 48.

$$-1,-48 -2,-24 -3,-16 -4,-12 -6,-8$$

Calculate the sum for each pair.

$$-1-48=-49 -2-24=-26 -3-16=-19 -4-12=-16 -6-8=-14$$

The solution is the pair that gives sum -26.

$$a=-24 b=-2$$

Rewrite x^{2}-26x+48 as \left(x^{2}-24x\right)+\left(-2x+48\right).

$$\left(x^{2}-24x\right)+\left(-2x+48\right)$$

Factor out x in the first and -2 in the second group.

$$x\left(x-24\right)-2\left(x-24\right)$$

Factor out common term x-24 by using distributive property.

$$\left(x-24\right)\left(x-2\right)$$

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.

$$x^{2}-26x+48=0$$

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.

$$x=\frac{-\left(-26\right)±\sqrt{\left(-26\right)^{2}-4\times 48}}{2}$$

Square -26.

$$x=\frac{-\left(-26\right)±\sqrt{676-4\times 48}}{2}$$

Multiply -4 times 48.

$$x=\frac{-\left(-26\right)±\sqrt{676-192}}{2}$$

Add 676 to -192.

$$x=\frac{-\left(-26\right)±\sqrt{484}}{2}$$

Take the square root of 484.

$$x=\frac{-\left(-26\right)±22}{2}$$

The opposite of -26 is 26.

$$x=\frac{26±22}{2}$$

Now solve the equation x=\frac{26±22}{2} when ± is plus. Add 26 to 22.

$$x=\frac{48}{2}$$

Divide 48 by 2.

$$x=24$$

Now solve the equation x=\frac{26±22}{2} when ± is minus. Subtract 22 from 26.

$$x=\frac{4}{2}$$

Divide 4 by 2.

$$x=2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 24 for x_{1} and 2 for x_{2}.

$$x^{2}-26x+48=\left(x-24\right)\left(x-2\right)$$

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.

$$x ^ 2 -26x +48 = 0$$

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

$$r + s = 26 rs = 48$$

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 = 48

$$(13 - u) (13 + u) = 48$$

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

$$169 - u^2 = 48$$

Simplify the expression by subtracting 169 on both sides

$$-u^2 = 48-169 = -121$$

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

$$u^2 = 121 u = \pm\sqrt{121} = \pm 11$$

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

$$r =13 - 11 = 2 s = 13 + 11 = 24$$

Evaluate

$$\left(x-24\right)\left(x-2\right)$$