Solve x^2+44x+363

$$x ^ { 2 } + 44 x + 363$$

Factor

$$\left(x+11\right)\left(x+33\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as x^{2}+ax+bx+363. To find a and b, set up a system to be solved.

$$a+b=44 ab=1\times 363=363$$

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 363.

$$1,363 3,121 11,33$$

Calculate the sum for each pair.

$$1+363=364 3+121=124 11+33=44$$

The solution is the pair that gives sum 44.

$$a=11 b=33$$

Rewrite x^{2}+44x+363 as \left(x^{2}+11x\right)+\left(33x+363\right).

$$\left(x^{2}+11x\right)+\left(33x+363\right)$$

Factor out x in the first and 33 in the second group.

$$x\left(x+11\right)+33\left(x+11\right)$$

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

$$\left(x+11\right)\left(x+33\right)$$

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.

$$x^{2}+44x+363=0$$

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.

$$x=\frac{-44±\sqrt{44^{2}-4\times 363}}{2}$$

Square 44.

$$x=\frac{-44±\sqrt{1936-4\times 363}}{2}$$

Multiply -4 times 363.

$$x=\frac{-44±\sqrt{1936-1452}}{2}$$

Add 1936 to -1452.

$$x=\frac{-44±\sqrt{484}}{2}$$

Take the square root of 484.

$$x=\frac{-44±22}{2}$$

Now solve the equation x=\frac{-44±22}{2} when ± is plus. Add -44 to 22.

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

Divide -22 by 2.

$$x=-11$$

Now solve the equation x=\frac{-44±22}{2} when ± is minus. Subtract 22 from -44.

$$x=-\frac{66}{2}$$

Divide -66 by 2.

$$x=-33$$

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 -33 for x_{2}.

$$x^{2}+44x+363=\left(x-\left(-11\right)\right)\left(x-\left(-33\right)\right)$$

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

$$x^{2}+44x+363=\left(x+11\right)\left(x+33\right)$$

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.

$$x ^ 2 +44x +363 = 0$$

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

$$r + s = -44 rs = 363$$

Two numbers r and s sum up to -44 exactly when the average of the two numbers is \frac{1}{2}*-44 = -22. 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>

$$r = -22 - u s = -22 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 363

$$(-22 - u) (-22 + u) = 363$$

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

$$484 - u^2 = 363$$

Simplify the expression by subtracting 484 on both sides

$$-u^2 = 363-484 = -121$$

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

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

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

$$r =-22 - 11 = -33 s = -22 + 11 = -11$$

Evaluate

$$\left(x+11\right)\left(x+33\right)$$