Solve 5b^2+12b+4

$$5 b ^ { 2 } + 12 b + 4$$

Factor

$$\left(b+2\right)\left(5b+2\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5b^{2}+pb+qb+4. To find p and q, set up a system to be solved.

$$p+q=12 pq=5\times 4=20$$

Since pq is positive, p and q have the same sign. Since p+q is positive, p and q are both positive. List all such integer pairs that give product 20.

$$1,20 2,10 4,5$$

Calculate the sum for each pair.

$$1+20=21 2+10=12 4+5=9$$

The solution is the pair that gives sum 12.

$$p=2 q=10$$

Rewrite 5b^{2}+12b+4 as \left(5b^{2}+2b\right)+\left(10b+4\right).

$$\left(5b^{2}+2b\right)+\left(10b+4\right)$$

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

$$b\left(5b+2\right)+2\left(5b+2\right)$$

Factor out common term 5b+2 by using distributive property.

$$\left(5b+2\right)\left(b+2\right)$$

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.

$$5b^{2}+12b+4=0$$

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.

$$b=\frac{-12±\sqrt{12^{2}-4\times 5\times 4}}{2\times 5}$$

Square 12.

$$b=\frac{-12±\sqrt{144-4\times 5\times 4}}{2\times 5}$$

Multiply -4 times 5.

$$b=\frac{-12±\sqrt{144-20\times 4}}{2\times 5}$$

Multiply -20 times 4.

$$b=\frac{-12±\sqrt{144-80}}{2\times 5}$$

Add 144 to -80.

$$b=\frac{-12±\sqrt{64}}{2\times 5}$$

Take the square root of 64.

$$b=\frac{-12±8}{2\times 5}$$

Multiply 2 times 5.

$$b=\frac{-12±8}{10}$$

Now solve the equation b=\frac{-12±8}{10} when ± is plus. Add -12 to 8.

$$b=-\frac{4}{10}$$

Reduce the fraction \frac{-4}{10} to lowest terms by extracting and canceling out 2.

$$b=-\frac{2}{5}$$

Now solve the equation b=\frac{-12±8}{10} when ± is minus. Subtract 8 from -12.

$$b=-\frac{20}{10}$$

Divide -20 by 10.

$$b=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{2}{5} for x_{1} and -2 for x_{2}.

$$5b^{2}+12b+4=5\left(b-\left(-\frac{2}{5}\right)\right)\left(b-\left(-2\right)\right)$$

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

$$5b^{2}+12b+4=5\left(b+\frac{2}{5}\right)\left(b+2\right)$$

Add \frac{2}{5} to b by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$5b^{2}+12b+4=5\times \frac{5b+2}{5}\left(b+2\right)$$

Cancel out 5, the greatest common factor in 5 and 5.

$$5b^{2}+12b+4=\left(5b+2\right)\left(b+2\right)$$

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 5

$$x ^ 2 +\frac{12}{5}x +\frac{4}{5} = 0$$

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

$$r + s = -\frac{12}{5} rs = \frac{4}{5}$$

Two numbers r and s sum up to -\frac{12}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{12}{5} = -\frac{6}{5}. 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>

$$r = -\frac{6}{5} - u s = -\frac{6}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{4}{5}

$$(-\frac{6}{5} - u) (-\frac{6}{5} + u) = \frac{4}{5}$$

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

$$\frac{36}{25} - u^2 = \frac{4}{5}$$

Simplify the expression by subtracting \frac{36}{25} on both sides

$$-u^2 = \frac{4}{5}-\frac{36}{25} = -\frac{16}{25}$$

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

$$u^2 = \frac{16}{25} u = \pm\sqrt{\frac{16}{25}} = \pm \frac{4}{5}$$

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

$$r =-\frac{6}{5} - \frac{4}{5} = -2 s = -\frac{6}{5} + \frac{4}{5} = -0.400$$

Evaluate

$$\left(b+2\right)\left(5b+2\right)$$