Solve -7v^2-25v-12

$$- 7 v ^ { 2 } - 25 v - 12$$

Factor

$$-\left(v+3\right)\left(7v+4\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as -7v^{2}+av+bv-12. To find a and b, set up a system to be solved.

$$a+b=-25 ab=-7\left(-12\right)=84$$

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 84.

$$-1,-84 -2,-42 -3,-28 -4,-21 -6,-14 -7,-12$$

Calculate the sum for each pair.

$$-1-84=-85 -2-42=-44 -3-28=-31 -4-21=-25 -6-14=-20 -7-12=-19$$

The solution is the pair that gives sum -25.

$$a=-4 b=-21$$

Rewrite -7v^{2}-25v-12 as \left(-7v^{2}-4v\right)+\left(-21v-12\right).

$$\left(-7v^{2}-4v\right)+\left(-21v-12\right)$$

Factor out -v in the first and -3 in the second group.

$$-v\left(7v+4\right)-3\left(7v+4\right)$$

Factor out common term 7v+4 by using distributive property.

$$\left(7v+4\right)\left(-v-3\right)$$

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.

$$-7v^{2}-25v-12=0$$

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.

$$v=\frac{-\left(-25\right)±\sqrt{\left(-25\right)^{2}-4\left(-7\right)\left(-12\right)}}{2\left(-7\right)}$$

Square -25.

$$v=\frac{-\left(-25\right)±\sqrt{625-4\left(-7\right)\left(-12\right)}}{2\left(-7\right)}$$

Multiply -4 times -7.

$$v=\frac{-\left(-25\right)±\sqrt{625+28\left(-12\right)}}{2\left(-7\right)}$$

Multiply 28 times -12.

$$v=\frac{-\left(-25\right)±\sqrt{625-336}}{2\left(-7\right)}$$

Add 625 to -336.

$$v=\frac{-\left(-25\right)±\sqrt{289}}{2\left(-7\right)}$$

Take the square root of 289.

$$v=\frac{-\left(-25\right)±17}{2\left(-7\right)}$$

The opposite of -25 is 25.

$$v=\frac{25±17}{2\left(-7\right)}$$

Multiply 2 times -7.

$$v=\frac{25±17}{-14}$$

Now solve the equation v=\frac{25±17}{-14} when ± is plus. Add 25 to 17.

$$v=\frac{42}{-14}$$

Divide 42 by -14.

$$v=-3$$

Now solve the equation v=\frac{25±17}{-14} when ± is minus. Subtract 17 from 25.

$$v=\frac{8}{-14}$$

Reduce the fraction \frac{8}{-14} to lowest terms by extracting and canceling out 2.

$$v=-\frac{4}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -3 for x_{1} and -\frac{4}{7} for x_{2}.

$$-7v^{2}-25v-12=-7\left(v-\left(-3\right)\right)\left(v-\left(-\frac{4}{7}\right)\right)$$

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

$$-7v^{2}-25v-12=-7\left(v+3\right)\left(v+\frac{4}{7}\right)$$

Add \frac{4}{7} to v by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$-7v^{2}-25v-12=-7\left(v+3\right)\times \frac{-7v-4}{-7}$$

Cancel out 7, the greatest common factor in -7 and 7.

$$-7v^{2}-25v-12=\left(v+3\right)\left(-7v-4\right)$$

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.

$$x ^ 2 +\frac{25}{7}x +\frac{12}{7} = 0$$

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

$$r + s = -\frac{25}{7} rs = \frac{12}{7}$$

Two numbers r and s sum up to -\frac{25}{7} exactly when the average of the two numbers is \frac{1}{2}*-\frac{25}{7} = -\frac{25}{14}. 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>

$$r = -\frac{25}{14} - u s = -\frac{25}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{12}{7}

$$(-\frac{25}{14} - u) (-\frac{25}{14} + u) = \frac{12}{7}$$

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

$$\frac{625}{196} - u^2 = \frac{12}{7}$$

Simplify the expression by subtracting \frac{625}{196} on both sides

$$-u^2 = \frac{12}{7}-\frac{625}{196} = -\frac{289}{196}$$

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

$$u^2 = \frac{289}{196} u = \pm\sqrt{\frac{289}{196}} = \pm \frac{17}{14}$$

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

$$r =-\frac{25}{14} - \frac{17}{14} = -3 s = -\frac{25}{14} + \frac{17}{14} = -0.571$$

Evaluate

$$-\left(v+3\right)\left(7v+4\right)$$