Solve 11f^2+13f+2

$$11 f ^ { 2 } + 13 f + 2$$

Factor

$$\left(f+1\right)\left(11f+2\right)$$

Solution Steps

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

$$a+b=13 ab=11\times 2=22$$

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 22.

$$1,22 2,11$$

Calculate the sum for each pair.

$$1+22=23 2+11=13$$

The solution is the pair that gives sum 13.

$$a=2 b=11$$

Rewrite 11f^{2}+13f+2 as \left(11f^{2}+2f\right)+\left(11f+2\right).

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

Factor out f in 11f^{2}+2f.

$$f\left(11f+2\right)+11f+2$$

Factor out common term 11f+2 by using distributive property.

$$\left(11f+2\right)\left(f+1\right)$$

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.

$$11f^{2}+13f+2=0$$

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.

$$f=\frac{-13±\sqrt{13^{2}-4\times 11\times 2}}{2\times 11}$$

Square 13.

$$f=\frac{-13±\sqrt{169-4\times 11\times 2}}{2\times 11}$$

Multiply -4 times 11.

$$f=\frac{-13±\sqrt{169-44\times 2}}{2\times 11}$$

Multiply -44 times 2.

$$f=\frac{-13±\sqrt{169-88}}{2\times 11}$$

Add 169 to -88.

$$f=\frac{-13±\sqrt{81}}{2\times 11}$$

Take the square root of 81.

$$f=\frac{-13±9}{2\times 11}$$

Multiply 2 times 11.

$$f=\frac{-13±9}{22}$$

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

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

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

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

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

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

Divide -22 by 22.

$$f=-1$$

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

$$11f^{2}+13f+2=11\left(f-\left(-\frac{2}{11}\right)\right)\left(f-\left(-1\right)\right)$$

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

$$11f^{2}+13f+2=11\left(f+\frac{2}{11}\right)\left(f+1\right)$$

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

$$11f^{2}+13f+2=11\times \frac{11f+2}{11}\left(f+1\right)$$

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

$$11f^{2}+13f+2=\left(11f+2\right)\left(f+1\right)$$

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

$$x ^ 2 +\frac{13}{11}x +\frac{2}{11} = 0$$

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

$$r + s = -\frac{13}{11} rs = \frac{2}{11}$$

Two numbers r and s sum up to -\frac{13}{11} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{11} = -\frac{13}{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>

$$r = -\frac{13}{22} - u s = -\frac{13}{22} + u$$

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

$$(-\frac{13}{22} - u) (-\frac{13}{22} + u) = \frac{2}{11}$$

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

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

Simplify the expression by subtracting \frac{169}{484} on both sides

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

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

$$u^2 = \frac{81}{484} u = \pm\sqrt{\frac{81}{484}} = \pm \frac{9}{22}$$

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

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

Evaluate

$$\left(f+1\right)\left(11f+2\right)$$