Solve 9x^2+20x+4

$$9 x ^ { 2 } + 20 x + 4$$

Factor

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

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 9x^{2}+ax+bx+4. To find a and b, set up a system to be solved.

$$a+b=20 ab=9\times 4=36$$

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 36.

$$1,36 2,18 3,12 4,9 6,6$$

Calculate the sum for each pair.

$$1+36=37 2+18=20 3+12=15 4+9=13 6+6=12$$

The solution is the pair that gives sum 20.

$$a=2 b=18$$

Rewrite 9x^{2}+20x+4 as \left(9x^{2}+2x\right)+\left(18x+4\right).

$$\left(9x^{2}+2x\right)+\left(18x+4\right)$$

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

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

Factor out common term 9x+2 by using distributive property.

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

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.

$$9x^{2}+20x+4=0$$

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.

$$x=\frac{-20±\sqrt{20^{2}-4\times 9\times 4}}{2\times 9}$$

Square 20.

$$x=\frac{-20±\sqrt{400-4\times 9\times 4}}{2\times 9}$$

Multiply -4 times 9.

$$x=\frac{-20±\sqrt{400-36\times 4}}{2\times 9}$$

Multiply -36 times 4.

$$x=\frac{-20±\sqrt{400-144}}{2\times 9}$$

Add 400 to -144.

$$x=\frac{-20±\sqrt{256}}{2\times 9}$$

Take the square root of 256.

$$x=\frac{-20±16}{2\times 9}$$

Multiply 2 times 9.

$$x=\frac{-20±16}{18}$$

Now solve the equation x=\frac{-20±16}{18} when ± is plus. Add -20 to 16.

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

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

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

Now solve the equation x=\frac{-20±16}{18} when ± is minus. Subtract 16 from -20.

$$x=-\frac{36}{18}$$

Divide -36 by 18.

$$x=-2$$

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

$$9x^{2}+20x+4=9\left(x-\left(-\frac{2}{9}\right)\right)\left(x-\left(-2\right)\right)$$

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

$$9x^{2}+20x+4=9\left(x+\frac{2}{9}\right)\left(x+2\right)$$

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

$$9x^{2}+20x+4=9\times \frac{9x+2}{9}\left(x+2\right)$$

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

$$9x^{2}+20x+4=\left(9x+2\right)\left(x+2\right)$$

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

$$x ^ 2 +\frac{20}{9}x +\frac{4}{9} = 0$$

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

$$r + s = -\frac{20}{9} rs = \frac{4}{9}$$

Two numbers r and s sum up to -\frac{20}{9} exactly when the average of the two numbers is \frac{1}{2}*-\frac{20}{9} = -\frac{10}{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>

$$r = -\frac{10}{9} - u s = -\frac{10}{9} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{9}

$$(-\frac{10}{9} - u) (-\frac{10}{9} + u) = \frac{4}{9}$$

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

$$\frac{100}{81} - u^2 = \frac{4}{9}$$

Simplify the expression by subtracting \frac{100}{81} on both sides

$$-u^2 = \frac{4}{9}-\frac{100}{81} = -\frac{64}{81}$$

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

$$u^2 = \frac{64}{81} u = \pm\sqrt{\frac{64}{81}} = \pm \frac{8}{9}$$

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

$$r =-\frac{10}{9} - \frac{8}{9} = -2 s = -\frac{10}{9} + \frac{8}{9} = -0.222$$

Evaluate

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