Solve 11x^2+2x-9

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

Factor

$$\left(11x-9\right)\left(x+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 11x^{2}+ax+bx-9. To find a and b, set up a system to be solved.

$$a+b=2 ab=11\left(-9\right)=-99$$

Since ab is negative, a and b have the opposite signs. Since a+b is positive, the positive number has greater absolute value than the negative. List all such integer pairs that give product -99.

$$-1,99 -3,33 -9,11$$

Calculate the sum for each pair.

$$-1+99=98 -3+33=30 -9+11=2$$

The solution is the pair that gives sum 2.

$$a=-9 b=11$$

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

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

Factor out x in 11x^{2}-9x.

$$x\left(11x-9\right)+11x-9$$

Factor out common term 11x-9 by using distributive property.

$$\left(11x-9\right)\left(x+1\right)$$

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.

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

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.

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

Square 2.

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

Multiply -4 times 11.

$$x=\frac{-2±\sqrt{4-44\left(-9\right)}}{2\times 11}$$

Multiply -44 times -9.

$$x=\frac{-2±\sqrt{4+396}}{2\times 11}$$

Add 4 to 396.

$$x=\frac{-2±\sqrt{400}}{2\times 11}$$

Take the square root of 400.

$$x=\frac{-2±20}{2\times 11}$$

Multiply 2 times 11.

$$x=\frac{-2±20}{22}$$

Now solve the equation x=\frac{-2±20}{22} when ± is plus. Add -2 to 20.

$$x=\frac{18}{22}$$

Reduce the fraction \frac{18}{22} to lowest terms by extracting and canceling out 2.

$$x=\frac{9}{11}$$

Now solve the equation x=\frac{-2±20}{22} when ± is minus. Subtract 20 from -2.

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

Divide -22 by 22.

$$x=-1$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{9}{11} for x_{1} and -1 for x_{2}.

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

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

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

Subtract \frac{9}{11} from x by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

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

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

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

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

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

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

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

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

$$r = -\frac{1}{11} - u s = -\frac{1}{11} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{9}{11}

$$(-\frac{1}{11} - u) (-\frac{1}{11} + u) = -\frac{9}{11}$$

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

$$\frac{1}{121} - u^2 = -\frac{9}{11}$$

Simplify the expression by subtracting \frac{1}{121} on both sides

$$-u^2 = -\frac{9}{11}-\frac{1}{121} = -\frac{100}{121}$$

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

$$u^2 = \frac{100}{121} u = \pm\sqrt{\frac{100}{121}} = \pm \frac{10}{11}$$

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

$$r =-\frac{1}{11} - \frac{10}{11} = -1 s = -\frac{1}{11} + \frac{10}{11} = 0.818$$

Evaluate

$$\left(11x-9\right)\left(x+1\right)$$