Solve 6v^2+17v+5

$$6 v ^ { 2 } + 17 v + 5$$

Factor

$$\left(2v+5\right)\left(3v+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 6v^{2}+av+bv+5. To find a and b, set up a system to be solved.

$$a+b=17 ab=6\times 5=30$$

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 30.

$$1,30 2,15 3,10 5,6$$

Calculate the sum for each pair.

$$1+30=31 2+15=17 3+10=13 5+6=11$$

The solution is the pair that gives sum 17.

$$a=2 b=15$$

Rewrite 6v^{2}+17v+5 as \left(6v^{2}+2v\right)+\left(15v+5\right).

$$\left(6v^{2}+2v\right)+\left(15v+5\right)$$

Factor out 2v in the first and 5 in the second group.

$$2v\left(3v+1\right)+5\left(3v+1\right)$$

Factor out common term 3v+1 by using distributive property.

$$\left(3v+1\right)\left(2v+5\right)$$

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.

$$6v^{2}+17v+5=0$$

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.

$$v=\frac{-17±\sqrt{17^{2}-4\times 6\times 5}}{2\times 6}$$

Square 17.

$$v=\frac{-17±\sqrt{289-4\times 6\times 5}}{2\times 6}$$

Multiply -4 times 6.

$$v=\frac{-17±\sqrt{289-24\times 5}}{2\times 6}$$

Multiply -24 times 5.

$$v=\frac{-17±\sqrt{289-120}}{2\times 6}$$

Add 289 to -120.

$$v=\frac{-17±\sqrt{169}}{2\times 6}$$

Take the square root of 169.

$$v=\frac{-17±13}{2\times 6}$$

Multiply 2 times 6.

$$v=\frac{-17±13}{12}$$

Now solve the equation v=\frac{-17±13}{12} when ± is plus. Add -17 to 13.

$$v=-\frac{4}{12}$$

Reduce the fraction \frac{-4}{12} to lowest terms by extracting and canceling out 4.

$$v=-\frac{1}{3}$$

Now solve the equation v=\frac{-17±13}{12} when ± is minus. Subtract 13 from -17.

$$v=-\frac{30}{12}$$

Reduce the fraction \frac{-30}{12} to lowest terms by extracting and canceling out 6.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{1}{3} for x_{1} and -\frac{5}{2} for x_{2}.

$$6v^{2}+17v+5=6\left(v-\left(-\frac{1}{3}\right)\right)\left(v-\left(-\frac{5}{2}\right)\right)$$

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

$$6v^{2}+17v+5=6\left(v+\frac{1}{3}\right)\left(v+\frac{5}{2}\right)$$

Add \frac{1}{3} to v by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$6v^{2}+17v+5=6\times \frac{3v+1}{3}\left(v+\frac{5}{2}\right)$$

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

$$6v^{2}+17v+5=6\times \frac{3v+1}{3}\times \frac{2v+5}{2}$$

Multiply \frac{3v+1}{3} times \frac{2v+5}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$6v^{2}+17v+5=6\times \frac{\left(3v+1\right)\left(2v+5\right)}{3\times 2}$$

Multiply 3 times 2.

$$6v^{2}+17v+5=6\times \frac{\left(3v+1\right)\left(2v+5\right)}{6}$$

Cancel out 6, the greatest common factor in 6 and 6.

$$6v^{2}+17v+5=\left(3v+1\right)\left(2v+5\right)$$

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 6

$$x ^ 2 +\frac{17}{6}x +\frac{5}{6} = 0$$

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

$$r + s = -\frac{17}{6} rs = \frac{5}{6}$$

Two numbers r and s sum up to -\frac{17}{6} exactly when the average of the two numbers is \frac{1}{2}*-\frac{17}{6} = -\frac{17}{12}. 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>

$$r = -\frac{17}{12} - u s = -\frac{17}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{6}

$$(-\frac{17}{12} - u) (-\frac{17}{12} + u) = \frac{5}{6}$$

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

$$\frac{289}{144} - u^2 = \frac{5}{6}$$

Simplify the expression by subtracting \frac{289}{144} on both sides

$$-u^2 = \frac{5}{6}-\frac{289}{144} = -\frac{169}{144}$$

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

$$u^2 = \frac{169}{144} u = \pm\sqrt{\frac{169}{144}} = \pm \frac{13}{12}$$

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

$$r =-\frac{17}{12} - \frac{13}{12} = -2.500 s = -\frac{17}{12} + \frac{13}{12} = -0.333$$

Evaluate

$$\left(2v+5\right)\left(3v+1\right)$$