Solve 24v^2+80v

$$24 v ^ { 2 } + 80 v$$

Factor

$$8v\left(3v+10\right)$$

Solution Steps

Factor out 8.

$$8\left(3v^{2}+10v\right)$$

Consider 3v^{2}+10v. Factor out v.

$$v\left(3v+10\right)$$

Rewrite the complete factored expression.

$$8v\left(3v+10\right)$$

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.

$$24v^{2}+80v=0$$

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.

$$v=\frac{-80±\sqrt{80^{2}}}{2\times 24}$$

Take the square root of 80^{2}.

$$v=\frac{-80±80}{2\times 24}$$

Multiply 2 times 24.

$$v=\frac{-80±80}{48}$$

Now solve the equation v=\frac{-80±80}{48} when ± is plus. Add -80 to 80.

$$v=\frac{0}{48}$$

Divide 0 by 48.

$$v=0$$

Now solve the equation v=\frac{-80±80}{48} when ± is minus. Subtract 80 from -80.

$$v=-\frac{160}{48}$$

Reduce the fraction \frac{-160}{48} to lowest terms by extracting and canceling out 16.

$$v=-\frac{10}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and -\frac{10}{3} for x_{2}.

$$24v^{2}+80v=24v\left(v-\left(-\frac{10}{3}\right)\right)$$

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

$$24v^{2}+80v=24v\left(v+\frac{10}{3}\right)$$

Add \frac{10}{3} to v by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$24v^{2}+80v=24v\times \frac{3v+10}{3}$$

Cancel out 3, the greatest common factor in 24 and 3.

$$24v^{2}+80v=8v\left(3v+10\right)$$

Evaluate

$$8v\left(3v+10\right)$$