Solve 3v^2-4v-7

$$3 v ^ { 2 } - 4 v - 7$$

Factor

$$\left(3v-7\right)\left(v+1\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 3v^{2}+av+bv-7. To find a and b, set up a system to be solved.

$$a+b=-4 ab=3\left(-7\right)=-21$$

Since ab is negative, a and b have the opposite signs. Since a+b is negative, the negative number has greater absolute value than the positive. List all such integer pairs that give product -21.

$$1,-21 3,-7$$

Calculate the sum for each pair.

$$1-21=-20 3-7=-4$$

The solution is the pair that gives sum -4.

$$a=-7 b=3$$

Rewrite 3v^{2}-4v-7 as \left(3v^{2}-7v\right)+\left(3v-7\right).

$$\left(3v^{2}-7v\right)+\left(3v-7\right)$$

Factor out v in 3v^{2}-7v.

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

Factor out common term 3v-7 by using distributive property.

$$\left(3v-7\right)\left(v+1\right)$$

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.

$$3v^{2}-4v-7=0$$

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.

$$v=\frac{-\left(-4\right)±\sqrt{\left(-4\right)^{2}-4\times 3\left(-7\right)}}{2\times 3}$$

Square -4.

$$v=\frac{-\left(-4\right)±\sqrt{16-4\times 3\left(-7\right)}}{2\times 3}$$

Multiply -4 times 3.

$$v=\frac{-\left(-4\right)±\sqrt{16-12\left(-7\right)}}{2\times 3}$$

Multiply -12 times -7.

$$v=\frac{-\left(-4\right)±\sqrt{16+84}}{2\times 3}$$

Add 16 to 84.

$$v=\frac{-\left(-4\right)±\sqrt{100}}{2\times 3}$$

Take the square root of 100.

$$v=\frac{-\left(-4\right)±10}{2\times 3}$$

The opposite of -4 is 4.

$$v=\frac{4±10}{2\times 3}$$

Multiply 2 times 3.

$$v=\frac{4±10}{6}$$

Now solve the equation v=\frac{4±10}{6} when ± is plus. Add 4 to 10.

$$v=\frac{14}{6}$$

Reduce the fraction \frac{14}{6} to lowest terms by extracting and canceling out 2.

$$v=\frac{7}{3}$$

Now solve the equation v=\frac{4±10}{6} when ± is minus. Subtract 10 from 4.

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

Divide -6 by 6.

$$v=-1$$

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

$$3v^{2}-4v-7=3\left(v-\frac{7}{3}\right)\left(v-\left(-1\right)\right)$$

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

$$3v^{2}-4v-7=3\left(v-\frac{7}{3}\right)\left(v+1\right)$$

Subtract \frac{7}{3} from v by finding a common denominator and subtracting the numerators. Then reduce the fraction to lowest terms if possible.

$$3v^{2}-4v-7=3\times \frac{3v-7}{3}\left(v+1\right)$$

Cancel out 3, the greatest common factor in 3 and 3.

$$3v^{2}-4v-7=\left(3v-7\right)\left(v+1\right)$$

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 3

$$x ^ 2 -\frac{4}{3}x -\frac{7}{3} = 0$$

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

$$r + s = \frac{4}{3} rs = -\frac{7}{3}$$

Two numbers r and s sum up to \frac{4}{3} exactly when the average of the two numbers is \frac{1}{2}*\frac{4}{3} = \frac{2}{3}. 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>

$$r = \frac{2}{3} - u s = \frac{2}{3} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{7}{3}

$$(\frac{2}{3} - u) (\frac{2}{3} + u) = -\frac{7}{3}$$

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

$$\frac{4}{9} - u^2 = -\frac{7}{3}$$

Simplify the expression by subtracting \frac{4}{9} on both sides

$$-u^2 = -\frac{7}{3}-\frac{4}{9} = -\frac{25}{9}$$

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

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

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

$$r =\frac{2}{3} - \frac{5}{3} = -1 s = \frac{2}{3} + \frac{5}{3} = 2.333$$

Evaluate

$$\left(3v-7\right)\left(v+1\right)$$