Solve 22x^2+127x+33

$$22 x ^ { 2 } + 127 x + 33$$

Factor

$$\left(2x+11\right)\left(11x+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 22x^{2}+ax+bx+33. To find a and b, set up a system to be solved.

$$a+b=127 ab=22\times 33=726$$

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 726.

$$1,726 2,363 3,242 6,121 11,66 22,33$$

Calculate the sum for each pair.

$$1+726=727 2+363=365 3+242=245 6+121=127 11+66=77 22+33=55$$

The solution is the pair that gives sum 127.

$$a=6 b=121$$

Rewrite 22x^{2}+127x+33 as \left(22x^{2}+6x\right)+\left(121x+33\right).

$$\left(22x^{2}+6x\right)+\left(121x+33\right)$$

Factor out 2x in the first and 11 in the second group.

$$2x\left(11x+3\right)+11\left(11x+3\right)$$

Factor out common term 11x+3 by using distributive property.

$$\left(11x+3\right)\left(2x+11\right)$$

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.

$$22x^{2}+127x+33=0$$

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.

$$x=\frac{-127±\sqrt{127^{2}-4\times 22\times 33}}{2\times 22}$$

Square 127.

$$x=\frac{-127±\sqrt{16129-4\times 22\times 33}}{2\times 22}$$

Multiply -4 times 22.

$$x=\frac{-127±\sqrt{16129-88\times 33}}{2\times 22}$$

Multiply -88 times 33.

$$x=\frac{-127±\sqrt{16129-2904}}{2\times 22}$$

Add 16129 to -2904.

$$x=\frac{-127±\sqrt{13225}}{2\times 22}$$

Take the square root of 13225.

$$x=\frac{-127±115}{2\times 22}$$

Multiply 2 times 22.

$$x=\frac{-127±115}{44}$$

Now solve the equation x=\frac{-127±115}{44} when ± is plus. Add -127 to 115.

$$x=-\frac{12}{44}$$

Reduce the fraction \frac{-12}{44} to lowest terms by extracting and canceling out 4.

$$x=-\frac{3}{11}$$

Now solve the equation x=\frac{-127±115}{44} when ± is minus. Subtract 115 from -127.

$$x=-\frac{242}{44}$$

Reduce the fraction \frac{-242}{44} to lowest terms by extracting and canceling out 22.

$$x=-\frac{11}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{11} for x_{1} and -\frac{11}{2} for x_{2}.

$$22x^{2}+127x+33=22\left(x-\left(-\frac{3}{11}\right)\right)\left(x-\left(-\frac{11}{2}\right)\right)$$

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

$$22x^{2}+127x+33=22\left(x+\frac{3}{11}\right)\left(x+\frac{11}{2}\right)$$

Add \frac{3}{11} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$22x^{2}+127x+33=22\times \frac{11x+3}{11}\left(x+\frac{11}{2}\right)$$

Add \frac{11}{2} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$22x^{2}+127x+33=22\times \frac{11x+3}{11}\times \frac{2x+11}{2}$$

Multiply \frac{11x+3}{11} times \frac{2x+11}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$22x^{2}+127x+33=22\times \frac{\left(11x+3\right)\left(2x+11\right)}{11\times 2}$$

Multiply 11 times 2.

$$22x^{2}+127x+33=22\times \frac{\left(11x+3\right)\left(2x+11\right)}{22}$$

Cancel out 22, the greatest common factor in 22 and 22.

$$22x^{2}+127x+33=\left(11x+3\right)\left(2x+11\right)$$

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 22

$$x ^ 2 +\frac{127}{22}x +\frac{3}{2} = 0$$

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

$$r + s = -\frac{127}{22} rs = \frac{3}{2}$$

Two numbers r and s sum up to -\frac{127}{22} exactly when the average of the two numbers is \frac{1}{2}*-\frac{127}{22} = -\frac{127}{44}. 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>

$$r = -\frac{127}{44} - u s = -\frac{127}{44} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{3}{2}

$$(-\frac{127}{44} - u) (-\frac{127}{44} + u) = \frac{3}{2}$$

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

$$\frac{16129}{1936} - u^2 = \frac{3}{2}$$

Simplify the expression by subtracting \frac{16129}{1936} on both sides

$$-u^2 = \frac{3}{2}-\frac{16129}{1936} = -\frac{13225}{1936}$$

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

$$u^2 = \frac{13225}{1936} u = \pm\sqrt{\frac{13225}{1936}} = \pm \frac{115}{44}$$

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

$$r =-\frac{127}{44} - \frac{115}{44} = -5.500 s = -\frac{127}{44} + \frac{115}{44} = -0.273$$

Evaluate

$$\left(2x+11\right)\left(11x+3\right)$$