Solve 5v^2+14v+8

$$5 v ^ { 2 } + 14 v + 8$$

Factor

$$\left(v+2\right)\left(5v+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 5v^{2}+av+bv+8. To find a and b, set up a system to be solved.

$$a+b=14 ab=5\times 8=40$$

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 40.

$$1,40 2,20 4,10 5,8$$

Calculate the sum for each pair.

$$1+40=41 2+20=22 4+10=14 5+8=13$$

The solution is the pair that gives sum 14.

$$a=4 b=10$$

Rewrite 5v^{2}+14v+8 as \left(5v^{2}+4v\right)+\left(10v+8\right).

$$\left(5v^{2}+4v\right)+\left(10v+8\right)$$

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

$$v\left(5v+4\right)+2\left(5v+4\right)$$

Factor out common term 5v+4 by using distributive property.

$$\left(5v+4\right)\left(v+2\right)$$

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.

$$5v^{2}+14v+8=0$$

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.

$$v=\frac{-14±\sqrt{14^{2}-4\times 5\times 8}}{2\times 5}$$

Square 14.

$$v=\frac{-14±\sqrt{196-4\times 5\times 8}}{2\times 5}$$

Multiply -4 times 5.

$$v=\frac{-14±\sqrt{196-20\times 8}}{2\times 5}$$

Multiply -20 times 8.

$$v=\frac{-14±\sqrt{196-160}}{2\times 5}$$

Add 196 to -160.

$$v=\frac{-14±\sqrt{36}}{2\times 5}$$

Take the square root of 36.

$$v=\frac{-14±6}{2\times 5}$$

Multiply 2 times 5.

$$v=\frac{-14±6}{10}$$

Now solve the equation v=\frac{-14±6}{10} when ± is plus. Add -14 to 6.

$$v=-\frac{8}{10}$$

Reduce the fraction \frac{-8}{10} to lowest terms by extracting and canceling out 2.

$$v=-\frac{4}{5}$$

Now solve the equation v=\frac{-14±6}{10} when ± is minus. Subtract 6 from -14.

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

Divide -20 by 10.

$$v=-2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{4}{5} for x_{1} and -2 for x_{2}.

$$5v^{2}+14v+8=5\left(v-\left(-\frac{4}{5}\right)\right)\left(v-\left(-2\right)\right)$$

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

$$5v^{2}+14v+8=5\left(v+\frac{4}{5}\right)\left(v+2\right)$$

Add \frac{4}{5} to v by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$5v^{2}+14v+8=5\times \frac{5v+4}{5}\left(v+2\right)$$

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

$$5v^{2}+14v+8=\left(5v+4\right)\left(v+2\right)$$

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

$$x ^ 2 +\frac{14}{5}x +\frac{8}{5} = 0$$

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

$$r + s = -\frac{14}{5} rs = \frac{8}{5}$$

Two numbers r and s sum up to -\frac{14}{5} exactly when the average of the two numbers is \frac{1}{2}*-\frac{14}{5} = -\frac{7}{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>

$$r = -\frac{7}{5} - u s = -\frac{7}{5} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{8}{5}

$$(-\frac{7}{5} - u) (-\frac{7}{5} + u) = \frac{8}{5}$$

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

$$\frac{49}{25} - u^2 = \frac{8}{5}$$

Simplify the expression by subtracting \frac{49}{25} on both sides

$$-u^2 = \frac{8}{5}-\frac{49}{25} = -\frac{9}{25}$$

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

$$u^2 = \frac{9}{25} u = \pm\sqrt{\frac{9}{25}} = \pm \frac{3}{5}$$

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

$$r =-\frac{7}{5} - \frac{3}{5} = -2.000 s = -\frac{7}{5} + \frac{3}{5} = -0.800$$

Evaluate

$$\left(v+2\right)\left(5v+4\right)$$