Solve x^2+22x+121=0

$$x ^ { 2 } + 22 x + 121 = 0$$

Solve for x

$$x=-11$$

Steps Using Factoring

To solve the equation, factor x^{2}+22x+121 using formula x^{2}+\left(a+b\right)x+ab=\left(x+a\right)\left(x+b\right). To find a and b, set up a system to be solved.

$$a+b=22 ab=121$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. 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 22.

$$a=11 b=11$$

Rewrite factored expression \left(x+a\right)\left(x+b\right) using the obtained values.

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

Rewrite as a binomial square.

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

To find equation solution, solve x+11=0.

$$x=-11$$

Steps Using Factoring By Grouping

To solve the equation, factor the left hand side by grouping. First, left hand side needs to be rewritten as x^{2}+ax+bx+121. To find a and b, set up a system to be solved.

$$a+b=22 ab=1\times 121=121$$

Since ab is positive, a and b have the same sign. Since a+b is positive, a and b are both positive. 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 22.

$$a=11 b=11$$

Rewrite x^{2}+22x+121 as \left(x^{2}+11x\right)+\left(11x+121\right).

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

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

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

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

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

Rewrite as a binomial square.

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

To find equation solution, solve x+11=0.

$$x=-11$$

Steps Using the Quadratic Formula

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.

$$x^{2}+22x+121=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 22 for b, and 121 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$x=\frac{-22±\sqrt{22^{2}-4\times 121}}{2}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 121}}{2}$$

Multiply -4 times 121.

$$x=\frac{-22±\sqrt{484-484}}{2}$$

Add 484 to -484.

$$x=\frac{-22±\sqrt{0}}{2}$$

Take the square root of 0.

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

Divide -22 by 2.

$$x=-11$$

Steps for Completing the Square

Factor x^{2}+22x+121. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(x+11\right)^{2}=0$$

Take the square root of both sides of the equation.

$$\sqrt{\left(x+11\right)^{2}}=\sqrt{0}$$

Simplify.

$$x+11=0 x+11=0$$

Subtract 11 from both sides of the equation.

$$x=-11 x=-11$$

The equation is now solved. Solutions are the same.

$$x=-11$$

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.

$$x ^ 2 +22x +121 = 0$$

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

$$r + s = -22 rs = 121$$

Two numbers r and s sum up to -22 exactly when the average of the two numbers is \frac{1}{2}*-22 = -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>

$$r = -11 - u s = -11 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 121

$$(-11 - u) (-11 + u) = 121$$

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

$$121 - u^2 = 121$$

Simplify the expression by subtracting 121 on both sides

$$-u^2 = 121-121 = 0$$

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

$$u^2 = 0 u = 0$$

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

$$r = s = -11$$