Solve 11x^2-122x+11

$$11 x ^ { 2 } - 122 x + 11$$

Factor

$$\left(x-11\right)\left(11x-1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 11x^{2}+ax+bx+11. To find a and b, set up a system to be solved.

$$a+b=-122 ab=11\times 11=121$$

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 121.

$$-1,-121 -11,-11$$

Calculate the sum for each pair.

$$-1-121=-122 -11-11=-22$$

The solution is the pair that gives sum -122.

$$a=-121 b=-1$$

Rewrite 11x^{2}-122x+11 as \left(11x^{2}-121x\right)+\left(-x+11\right).

$$\left(11x^{2}-121x\right)+\left(-x+11\right)$$

Factor out 11x in the first and -1 in the second group.

$$11x\left(x-11\right)-\left(x-11\right)$$

Factor out common term x-11 by using distributive property.

$$\left(x-11\right)\left(11x-1\right)$$

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.

$$11x^{2}-122x+11=0$$

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.

$$x=\frac{-\left(-122\right)±\sqrt{\left(-122\right)^{2}-4\times 11\times 11}}{2\times 11}$$

Square -122.

$$x=\frac{-\left(-122\right)±\sqrt{14884-4\times 11\times 11}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-122\right)±\sqrt{14884-44\times 11}}{2\times 11}$$

Multiply -44 times 11.

$$x=\frac{-\left(-122\right)±\sqrt{14884-484}}{2\times 11}$$

Add 14884 to -484.

$$x=\frac{-\left(-122\right)±\sqrt{14400}}{2\times 11}$$

Take the square root of 14400.

$$x=\frac{-\left(-122\right)±120}{2\times 11}$$

The opposite of -122 is 122.

$$x=\frac{122±120}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{122±120}{22}$$

Now solve the equation x=\frac{122±120}{22} when ± is plus. Add 122 to 120.

$$x=\frac{242}{22}$$

Divide 242 by 22.

$$x=11$$

Now solve the equation x=\frac{122±120}{22} when ± is minus. Subtract 120 from 122.

$$x=\frac{2}{22}$$

Reduce the fraction \frac{2}{22} to lowest terms by extracting and canceling out 2.

$$x=\frac{1}{11}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 11 for x_{1} and \frac{1}{11} for x_{2}.

$$11x^{2}-122x+11=11\left(x-11\right)\left(x-\frac{1}{11}\right)$$

Subtract \frac{1}{11} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$11x^{2}-122x+11=11\left(x-11\right)\times \frac{11x-1}{11}$$

Cancel out 11, the greatest common factor in 11 and 11.

$$11x^{2}-122x+11=\left(x-11\right)\left(11x-1\right)$$

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.This is achieved by dividing both sides of the equation by 11

$$x ^ 2 -\frac{122}{11}x +1 = 0$$

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

$$r + s = \frac{122}{11} rs = 1$$

Two numbers r and s sum up to \frac{122}{11} exactly when the average of the two numbers is \frac{1}{2}*\frac{122}{11} = \frac{61}{11}. 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>

$$r = \frac{61}{11} - u s = \frac{61}{11} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 1

$$(\frac{61}{11} - u) (\frac{61}{11} + u) = 1$$

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

$$\frac{3721}{121} - u^2 = 1$$

Simplify the expression by subtracting \frac{3721}{121} on both sides

$$-u^2 = 1-\frac{3721}{121} = -\frac{3600}{121}$$

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

$$u^2 = \frac{3600}{121} u = \pm\sqrt{\frac{3600}{121}} = \pm \frac{60}{11}$$

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

$$r =\frac{61}{11} - \frac{60}{11} = 0.091 s = \frac{61}{11} + \frac{60}{11} = 11$$

Evaluate

$$\left(x-11\right)\left(11x-1\right)$$