Solve x^2+11x-26

$$x ^ { 2 } + 11 x - 26$$

Factor

$$\left(x-2\right)\left(x+13\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx-26. To find a and b, set up a system to be solved.

$$a+b=11 ab=1\left(-26\right)=-26$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -26.

$$-1,26 -2,13$$

Calculate the sum for each pair.

$$-1+26=25 -2+13=11$$

The solution is the pair that gives sum 11.

$$a=-2 b=13$$

Rewrite x^{2}+11x-26 as \left(x^{2}-2x\right)+\left(13x-26\right).

$$\left(x^{2}-2x\right)+\left(13x-26\right)$$

Factor out x in the first and 13 in the second group.

$$x\left(x-2\right)+13\left(x-2\right)$$

Factor out common term x-2 by using distributive property.

$$\left(x-2\right)\left(x+13\right)$$

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.

$$x^{2}+11x-26=0$$

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.

$$x=\frac{-11±\sqrt{11^{2}-4\left(-26\right)}}{2}$$

Square 11.

$$x=\frac{-11±\sqrt{121-4\left(-26\right)}}{2}$$

Multiply -4 times -26.

$$x=\frac{-11±\sqrt{121+104}}{2}$$

Add 121 to 104.

$$x=\frac{-11±\sqrt{225}}{2}$$

Take the square root of 225.

$$x=\frac{-11±15}{2}$$

Now solve the equation x=\frac{-11±15}{2} when ± is plus. Add -11 to 15.

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

Divide 4 by 2.

$$x=2$$

Now solve the equation x=\frac{-11±15}{2} when ± is minus. Subtract 15 from -11.

$$x=-\frac{26}{2}$$

Divide -26 by 2.

$$x=-13$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2 for x_{1} and -13 for x_{2}.

$$x^{2}+11x-26=\left(x-2\right)\left(x-\left(-13\right)\right)$$

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

$$x^{2}+11x-26=\left(x-2\right)\left(x+13\right)$$

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.

$$x ^ 2 +11x -26 = 0$$

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

$$r + s = -11 rs = -26$$

Two numbers r and s sum up to -11 exactly when the average of the two numbers is \frac{1}{2}*-11 = -\frac{11}{2}. 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>

$$r = -\frac{11}{2} - u s = -\frac{11}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -26

$$(-\frac{11}{2} - u) (-\frac{11}{2} + u) = -26$$

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

$$\frac{121}{4} - u^2 = -26$$

Simplify the expression by subtracting \frac{121}{4} on both sides

$$-u^2 = -26-\frac{121}{4} = -\frac{225}{4}$$

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

$$u^2 = \frac{225}{4} u = \pm\sqrt{\frac{225}{4}} = \pm \frac{15}{2}$$

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

$$r =-\frac{11}{2} - \frac{15}{2} = -13 s = -\frac{11}{2} + \frac{15}{2} = 2$$

Evaluate

$$\left(x-2\right)\left(x+13\right)$$