Solve 11x^2-9x-2

$$11 x ^ { 2 } - 9 x - 2$$

Factor

$$\left(x-1\right)\left(11x+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 11x^{2}+ax+bx-2. To find a and b, set up a system to be solved.

$$a+b=-9 ab=11\left(-2\right)=-22$$

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 -22.

$$1,-22 2,-11$$

Calculate the sum for each pair.

$$1-22=-21 2-11=-9$$

The solution is the pair that gives sum -9.

$$a=-11 b=2$$

Rewrite 11x^{2}-9x-2 as \left(11x^{2}-11x\right)+\left(2x-2\right).

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

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

$$11x\left(x-1\right)+2\left(x-1\right)$$

Factor out common term x-1 by using distributive property.

$$\left(x-1\right)\left(11x+2\right)$$

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.

$$11x^{2}-9x-2=0$$

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.

$$x=\frac{-\left(-9\right)±\sqrt{\left(-9\right)^{2}-4\times 11\left(-2\right)}}{2\times 11}$$

Square -9.

$$x=\frac{-\left(-9\right)±\sqrt{81-4\times 11\left(-2\right)}}{2\times 11}$$

Multiply -4 times 11.

$$x=\frac{-\left(-9\right)±\sqrt{81-44\left(-2\right)}}{2\times 11}$$

Multiply -44 times -2.

$$x=\frac{-\left(-9\right)±\sqrt{81+88}}{2\times 11}$$

Add 81 to 88.

$$x=\frac{-\left(-9\right)±\sqrt{169}}{2\times 11}$$

Take the square root of 169.

$$x=\frac{-\left(-9\right)±13}{2\times 11}$$

The opposite of -9 is 9.

$$x=\frac{9±13}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{9±13}{22}$$

Now solve the equation x=\frac{9±13}{22} when ± is plus. Add 9 to 13.

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

Divide 22 by 22.

$$x=1$$

Now solve the equation x=\frac{9±13}{22} when ± is minus. Subtract 13 from 9.

$$x=-\frac{4}{22}$$

Reduce the fraction \frac{-4}{22} to lowest terms by extracting and canceling out 2.

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

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

$$11x^{2}-9x-2=11\left(x-1\right)\left(x-\left(-\frac{2}{11}\right)\right)$$

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

$$11x^{2}-9x-2=11\left(x-1\right)\left(x+\frac{2}{11}\right)$$

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

$$11x^{2}-9x-2=11\left(x-1\right)\times \frac{11x+2}{11}$$

Cancel out 11, the greatest common factor in 11 and 11.

$$11x^{2}-9x-2=\left(x-1\right)\left(11x+2\right)$$

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 11

$$x ^ 2 -\frac{9}{11}x -\frac{2}{11} = 0$$

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

$$r + s = \frac{9}{11} rs = -\frac{2}{11}$$

Two numbers r and s sum up to \frac{9}{11} exactly when the average of the two numbers is \frac{1}{2}*\frac{9}{11} = \frac{9}{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>

$$r = \frac{9}{22} - u s = \frac{9}{22} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{2}{11}

$$(\frac{9}{22} - u) (\frac{9}{22} + u) = -\frac{2}{11}$$

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

$$\frac{81}{484} - u^2 = -\frac{2}{11}$$

Simplify the expression by subtracting \frac{81}{484} on both sides

$$-u^2 = -\frac{2}{11}-\frac{81}{484} = -\frac{169}{484}$$

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

$$u^2 = \frac{169}{484} u = \pm\sqrt{\frac{169}{484}} = \pm \frac{13}{22}$$

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

$$r =\frac{9}{22} - \frac{13}{22} = -0.182 s = \frac{9}{22} + \frac{13}{22} = 1$$

Evaluate

$$\left(x-1\right)\left(11x+2\right)$$