Solve x^2+22x+40

$$x ^ { 2 } + 22 x + 40$$

Factor

$$\left(x+2\right)\left(x+20\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+40. To find a and b, set up a system to be solved.

$$a+b=22 ab=1\times 40=40$$

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 40.

$$1,40 2,20 4,10 5,8$$

Calculate the sum for each pair.

$$1+40=41 2+20=22 4+10=14 5+8=13$$

The solution is the pair that gives sum 22.

$$a=2 b=20$$

Rewrite x^{2}+22x+40 as \left(x^{2}+2x\right)+\left(20x+40\right).

$$\left(x^{2}+2x\right)+\left(20x+40\right)$$

Factor out x in the first and 20 in the second group.

$$x\left(x+2\right)+20\left(x+2\right)$$

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

$$\left(x+2\right)\left(x+20\right)$$

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.

$$x^{2}+22x+40=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\times 40}}{2}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 40}}{2}$$

Multiply -4 times 40.

$$x=\frac{-22±\sqrt{484-160}}{2}$$

Add 484 to -160.

$$x=\frac{-22±\sqrt{324}}{2}$$

Take the square root of 324.

$$x=\frac{-22±18}{2}$$

Now solve the equation x=\frac{-22±18}{2} when ± is plus. Add -22 to 18.

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

Divide -4 by 2.

$$x=-2$$

Now solve the equation x=\frac{-22±18}{2} when ± is minus. Subtract 18 from -22.

$$x=-\frac{40}{2}$$

Divide -40 by 2.

$$x=-20$$

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 -20 for x_{2}.

$$x^{2}+22x+40=\left(x-\left(-2\right)\right)\left(x-\left(-20\right)\right)$$

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

$$x^{2}+22x+40=\left(x+2\right)\left(x+20\right)$$

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.

$$x ^ 2 +22x +40 = 0$$

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

$$r + s = -22 rs = 40$$

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 = 40

$$(-11 - u) (-11 + u) = 40$$

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

$$121 - u^2 = 40$$

Simplify the expression by subtracting 121 on both sides

$$-u^2 = 40-121 = -81$$

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

$$u^2 = 81 u = \pm\sqrt{81} = \pm 9$$

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

$$r =-11 - 9 = -20 s = -11 + 9 = -2$$

Evaluate

$$\left(x+2\right)\left(x+20\right)$$