Solve 9x^2-196x-44

$$9 x ^ { 2 } - 196 x - 44$$

Factor

$$\left(x-22\right)\left(9x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9x^{2}+ax+bx-44. To find a and b, set up a system to be solved.

$$a+b=-196 ab=9\left(-44\right)=-396$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -396.

$$1,-396 2,-198 3,-132 4,-99 6,-66 9,-44 11,-36 12,-33 18,-22$$

Calculate the sum for each pair.

$$1-396=-395 2-198=-196 3-132=-129 4-99=-95 6-66=-60 9-44=-35 11-36=-25 12-33=-21 18-22=-4$$

The solution is the pair that gives sum -196.

$$a=-198 b=2$$

Rewrite 9x^{2}-196x-44 as \left(9x^{2}-198x\right)+\left(2x-44\right).

$$\left(9x^{2}-198x\right)+\left(2x-44\right)$$

Factor out 9x in the first and 2 in the second group.

$$9x\left(x-22\right)+2\left(x-22\right)$$

Factor out common term x-22 by using distributive property.

$$\left(x-22\right)\left(9x+2\right)$$

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.

$$9x^{2}-196x-44=0$$

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.

$$x=\frac{-\left(-196\right)±\sqrt{\left(-196\right)^{2}-4\times 9\left(-44\right)}}{2\times 9}$$

Square -196.

$$x=\frac{-\left(-196\right)±\sqrt{38416-4\times 9\left(-44\right)}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-\left(-196\right)±\sqrt{38416-36\left(-44\right)}}{2\times 9}$$

Multiply -36 times -44.

$$x=\frac{-\left(-196\right)±\sqrt{38416+1584}}{2\times 9}$$

Add 38416 to 1584.

$$x=\frac{-\left(-196\right)±\sqrt{40000}}{2\times 9}$$

Take the square root of 40000.

$$x=\frac{-\left(-196\right)±200}{2\times 9}$$

The opposite of -196 is 196.

$$x=\frac{196±200}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{196±200}{18}$$

Now solve the equation x=\frac{196±200}{18} when ± is plus. Add 196 to 200.

$$x=\frac{396}{18}$$

Divide 396 by 18.

$$x=22$$

Now solve the equation x=\frac{196±200}{18} when ± is minus. Subtract 200 from 196.

$$x=-\frac{4}{18}$$

Reduce the fraction \frac{-4}{18} to lowest terms by extracting and canceling out 2.

$$x=-\frac{2}{9}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 22 for x_{1} and -\frac{2}{9} for x_{2}.

$$9x^{2}-196x-44=9\left(x-22\right)\left(x-\left(-\frac{2}{9}\right)\right)$$

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

$$9x^{2}-196x-44=9\left(x-22\right)\left(x+\frac{2}{9}\right)$$

Add \frac{2}{9} to x by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$9x^{2}-196x-44=9\left(x-22\right)\times \frac{9x+2}{9}$$

Cancel out 9, the greatest common factor in 9 and 9.

$$9x^{2}-196x-44=\left(x-22\right)\left(9x+2\right)$$

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 9

$$x ^ 2 -\frac{196}{9}x -\frac{44}{9} = 0$$

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

$$r + s = \frac{196}{9} rs = -\frac{44}{9}$$

Two numbers r and s sum up to \frac{196}{9} exactly when the average of the two numbers is \frac{1}{2}*\frac{196}{9} = \frac{98}{9}. 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>

$$r = \frac{98}{9} - u s = \frac{98}{9} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{44}{9}

$$(\frac{98}{9} - u) (\frac{98}{9} + u) = -\frac{44}{9}$$

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

$$\frac{9604}{81} - u^2 = -\frac{44}{9}$$

Simplify the expression by subtracting \frac{9604}{81} on both sides

$$-u^2 = -\frac{44}{9}-\frac{9604}{81} = -\frac{10000}{81}$$

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

$$u^2 = \frac{10000}{81} u = \pm\sqrt{\frac{10000}{81}} = \pm \frac{100}{9}$$

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

$$r =\frac{98}{9} - \frac{100}{9} = -0.222 s = \frac{98}{9} + \frac{100}{9} = 22$$

Evaluate

$$\left(x-22\right)\left(9x+2\right)$$