Solve x^2+22x+85

$$x ^ { 2 } + 22 x + 85$$

Factor

$$\left(x+5\right)\left(x+17\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+85. To find a and b, set up a system to be solved.

$$a+b=22 ab=1\times 85=85$$

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 85.

$$1,85 5,17$$

Calculate the sum for each pair.

$$1+85=86 5+17=22$$

The solution is the pair that gives sum 22.

$$a=5 b=17$$

Rewrite x^{2}+22x+85 as \left(x^{2}+5x\right)+\left(17x+85\right).

$$\left(x^{2}+5x\right)+\left(17x+85\right)$$

Factor out x in the first and 17 in the second group.

$$x\left(x+5\right)+17\left(x+5\right)$$

Factor out common term x+5 by using distributive property.

$$\left(x+5\right)\left(x+17\right)$$

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.

$$x^{2}+22x+85=0$$

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.

$$x=\frac{-22±\sqrt{22^{2}-4\times 85}}{2}$$

Square 22.

$$x=\frac{-22±\sqrt{484-4\times 85}}{2}$$

Multiply -4 times 85.

$$x=\frac{-22±\sqrt{484-340}}{2}$$

Add 484 to -340.

$$x=\frac{-22±\sqrt{144}}{2}$$

Take the square root of 144.

$$x=\frac{-22±12}{2}$$

Now solve the equation x=\frac{-22±12}{2} when ± is plus. Add -22 to 12.

$$x=-\frac{10}{2}$$

Divide -10 by 2.

$$x=-5$$

Now solve the equation x=\frac{-22±12}{2} when ± is minus. Subtract 12 from -22.

$$x=-\frac{34}{2}$$

Divide -34 by 2.

$$x=-17$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -5 for x_{1} and -17 for x_{2}.

$$x^{2}+22x+85=\left(x-\left(-5\right)\right)\left(x-\left(-17\right)\right)$$

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

$$x^{2}+22x+85=\left(x+5\right)\left(x+17\right)$$

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.

$$x ^ 2 +22x +85 = 0$$

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

$$r + s = -22 rs = 85$$

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 = 85

$$(-11 - u) (-11 + u) = 85$$

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

$$121 - u^2 = 85$$

Simplify the expression by subtracting 121 on both sides

$$-u^2 = 85-121 = -36$$

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

$$u^2 = 36 u = \pm\sqrt{36} = \pm 6$$

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

$$r =-11 - 6 = -17 s = -11 + 6 = -5$$

Evaluate

$$\left(x+5\right)\left(x+17\right)$$