Solve 8v^2+26v+15

$$8 v ^ { 2 } + 26 v + 15$$

Factor

$$\left(2v+5\right)\left(4v+3\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 8v^{2}+av+bv+15. To find a and b, set up a system to be solved.

$$a+b=26 ab=8\times 15=120$$

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 120.

$$1,120 2,60 3,40 4,30 5,24 6,20 8,15 10,12$$

Calculate the sum for each pair.

$$1+120=121 2+60=62 3+40=43 4+30=34 5+24=29 6+20=26 8+15=23 10+12=22$$

The solution is the pair that gives sum 26.

$$a=6 b=20$$

Rewrite 8v^{2}+26v+15 as \left(8v^{2}+6v\right)+\left(20v+15\right).

$$\left(8v^{2}+6v\right)+\left(20v+15\right)$$

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

$$2v\left(4v+3\right)+5\left(4v+3\right)$$

Factor out common term 4v+3 by using distributive property.

$$\left(4v+3\right)\left(2v+5\right)$$

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.

$$8v^{2}+26v+15=0$$

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.

$$v=\frac{-26±\sqrt{26^{2}-4\times 8\times 15}}{2\times 8}$$

Square 26.

$$v=\frac{-26±\sqrt{676-4\times 8\times 15}}{2\times 8}$$

Multiply -4 times 8.

$$v=\frac{-26±\sqrt{676-32\times 15}}{2\times 8}$$

Multiply -32 times 15.

$$v=\frac{-26±\sqrt{676-480}}{2\times 8}$$

Add 676 to -480.

$$v=\frac{-26±\sqrt{196}}{2\times 8}$$

Take the square root of 196.

$$v=\frac{-26±14}{2\times 8}$$

Multiply 2 times 8.

$$v=\frac{-26±14}{16}$$

Now solve the equation v=\frac{-26±14}{16} when ± is plus. Add -26 to 14.

$$v=-\frac{12}{16}$$

Reduce the fraction \frac{-12}{16} to lowest terms by extracting and canceling out 4.

$$v=-\frac{3}{4}$$

Now solve the equation v=\frac{-26±14}{16} when ± is minus. Subtract 14 from -26.

$$v=-\frac{40}{16}$$

Reduce the fraction \frac{-40}{16} to lowest terms by extracting and canceling out 8.

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

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{3}{4} for x_{1} and -\frac{5}{2} for x_{2}.

$$8v^{2}+26v+15=8\left(v-\left(-\frac{3}{4}\right)\right)\left(v-\left(-\frac{5}{2}\right)\right)$$

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

$$8v^{2}+26v+15=8\left(v+\frac{3}{4}\right)\left(v+\frac{5}{2}\right)$$

Add \frac{3}{4} to v by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$8v^{2}+26v+15=8\times \frac{4v+3}{4}\left(v+\frac{5}{2}\right)$$

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

$$8v^{2}+26v+15=8\times \frac{4v+3}{4}\times \frac{2v+5}{2}$$

Multiply \frac{4v+3}{4} times \frac{2v+5}{2} by multiplying numerator times numerator and denominator times denominator. Then reduce the fraction to lowest terms if possible.

$$8v^{2}+26v+15=8\times \frac{\left(4v+3\right)\left(2v+5\right)}{4\times 2}$$

Multiply 4 times 2.

$$8v^{2}+26v+15=8\times \frac{\left(4v+3\right)\left(2v+5\right)}{8}$$

Cancel out 8, the greatest common factor in 8 and 8.

$$8v^{2}+26v+15=\left(4v+3\right)\left(2v+5\right)$$

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 8

$$x ^ 2 +\frac{13}{4}x +\frac{15}{8} = 0$$

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

$$r + s = -\frac{13}{4} rs = \frac{15}{8}$$

Two numbers r and s sum up to -\frac{13}{4} exactly when the average of the two numbers is \frac{1}{2}*-\frac{13}{4} = -\frac{13}{8}. 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>

$$r = -\frac{13}{8} - u s = -\frac{13}{8} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{15}{8}

$$(-\frac{13}{8} - u) (-\frac{13}{8} + u) = \frac{15}{8}$$

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

$$\frac{169}{64} - u^2 = \frac{15}{8}$$

Simplify the expression by subtracting \frac{169}{64} on both sides

$$-u^2 = \frac{15}{8}-\frac{169}{64} = -\frac{49}{64}$$

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

$$u^2 = \frac{49}{64} u = \pm\sqrt{\frac{49}{64}} = \pm \frac{7}{8}$$

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

$$r =-\frac{13}{8} - \frac{7}{8} = -2.500 s = -\frac{13}{8} + \frac{7}{8} = -0.750$$

Evaluate

$$\left(2v+5\right)\left(4v+3\right)$$