Solve -2v^2-13v-15

$$- 2 v ^ { 2 } - 13 v - 15$$

Factor

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

Solution Steps

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

$$a+b=-13 ab=-2\left(-15\right)=30$$

Since ab is positive, a and b have the same sign. Since a+b is negative, a and b are both negative. 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 -13.

$$a=-3 b=-10$$

Rewrite -2v^{2}-13v-15 as \left(-2v^{2}-3v\right)+\left(-10v-15\right).

$$\left(-2v^{2}-3v\right)+\left(-10v-15\right)$$

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

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

Factor out common term 2v+3 by using distributive property.

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

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.

$$-2v^{2}-13v-15=0$$

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.

$$v=\frac{-\left(-13\right)±\sqrt{\left(-13\right)^{2}-4\left(-2\right)\left(-15\right)}}{2\left(-2\right)}$$

Square -13.

$$v=\frac{-\left(-13\right)±\sqrt{169-4\left(-2\right)\left(-15\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$v=\frac{-\left(-13\right)±\sqrt{169+8\left(-15\right)}}{2\left(-2\right)}$$

Multiply 8 times -15.

$$v=\frac{-\left(-13\right)±\sqrt{169-120}}{2\left(-2\right)}$$

Add 169 to -120.

$$v=\frac{-\left(-13\right)±\sqrt{49}}{2\left(-2\right)}$$

Take the square root of 49.

$$v=\frac{-\left(-13\right)±7}{2\left(-2\right)}$$

The opposite of -13 is 13.

$$v=\frac{13±7}{2\left(-2\right)}$$

Multiply 2 times -2.

$$v=\frac{13±7}{-4}$$

Now solve the equation v=\frac{13±7}{-4} when ± is plus. Add 13 to 7.

$$v=\frac{20}{-4}$$

Divide 20 by -4.

$$v=-5$$

Now solve the equation v=\frac{13±7}{-4} when ± is minus. Subtract 7 from 13.

$$v=\frac{6}{-4}$$

Reduce the fraction \frac{6}{-4} to lowest terms by extracting and canceling out 2.

$$v=-\frac{3}{2}$$

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

$$-2v^{2}-13v-15=-2\left(v-\left(-5\right)\right)\left(v-\left(-\frac{3}{2}\right)\right)$$

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

$$-2v^{2}-13v-15=-2\left(v+5\right)\left(v+\frac{3}{2}\right)$$

Add \frac{3}{2} to v by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-2v^{2}-13v-15=-2\left(v+5\right)\times \frac{-2v-3}{-2}$$

Cancel out 2, the greatest common factor in -2 and 2.

$$-2v^{2}-13v-15=\left(v+5\right)\left(-2v-3\right)$$

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.

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

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

$$r + s = -\frac{13}{2} rs = \frac{15}{2}$$

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

$$r = -\frac{13}{4} - u s = -\frac{13}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{15}{2}

$$(-\frac{13}{4} - u) (-\frac{13}{4} + u) = \frac{15}{2}$$

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

$$\frac{169}{16} - u^2 = \frac{15}{2}$$

Simplify the expression by subtracting \frac{169}{16} on both sides

$$-u^2 = \frac{15}{2}-\frac{169}{16} = -\frac{49}{16}$$

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

$$u^2 = \frac{49}{16} u = \pm\sqrt{\frac{49}{16}} = \pm \frac{7}{4}$$

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

$$r =-\frac{13}{4} - \frac{7}{4} = -5 s = -\frac{13}{4} + \frac{7}{4} = -1.500$$

Evaluate

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