Solve 4v^2-4-15v

$$4 v ^ { 2 } - 4 - 15 v$$

Factor

$$\left(v-4\right)\left(4v+1\right)$$

Solution Steps

Rearrange the polynomial to put it in standard form. Place the terms in order from highest to lowest power.

$$4v^{2}-15v-4$$

Factor the expression by grouping. First, the expression needs to be rewritten as 4v^{2}+av+bv-4. To find a and b, set up a system to be solved.

$$a+b=-15 ab=4\left(-4\right)=-16$$

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 -16.

$$1,-16 2,-8 4,-4$$

Calculate the sum for each pair.

$$1-16=-15 2-8=-6 4-4=0$$

The solution is the pair that gives sum -15.

$$a=-16 b=1$$

Rewrite 4v^{2}-15v-4 as \left(4v^{2}-16v\right)+\left(v-4\right).

$$\left(4v^{2}-16v\right)+\left(v-4\right)$$

Factor out 4v in 4v^{2}-16v.

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

Factor out common term v-4 by using distributive property.

$$\left(v-4\right)\left(4v+1\right)$$

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.

$$4v^{2}-15v-4=0$$

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.

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

Square -15.

$$v=\frac{-\left(-15\right)±\sqrt{225-4\times 4\left(-4\right)}}{2\times 4}$$

Multiply -4 times 4.

$$v=\frac{-\left(-15\right)±\sqrt{225-16\left(-4\right)}}{2\times 4}$$

Multiply -16 times -4.

$$v=\frac{-\left(-15\right)±\sqrt{225+64}}{2\times 4}$$

Add 225 to 64.

$$v=\frac{-\left(-15\right)±\sqrt{289}}{2\times 4}$$

Take the square root of 289.

$$v=\frac{-\left(-15\right)±17}{2\times 4}$$

The opposite of -15 is 15.

$$v=\frac{15±17}{2\times 4}$$

Multiply 2 times 4.

$$v=\frac{15±17}{8}$$

Now solve the equation v=\frac{15±17}{8} when ± is plus. Add 15 to 17.

$$v=\frac{32}{8}$$

Divide 32 by 8.

$$v=4$$

Now solve the equation v=\frac{15±17}{8} when ± is minus. Subtract 17 from 15.

$$v=-\frac{2}{8}$$

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

$$v=-\frac{1}{4}$$

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

$$4v^{2}-15v-4=4\left(v-4\right)\left(v-\left(-\frac{1}{4}\right)\right)$$

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

$$4v^{2}-15v-4=4\left(v-4\right)\left(v+\frac{1}{4}\right)$$

Add \frac{1}{4} to v by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$4v^{2}-15v-4=4\left(v-4\right)\times \frac{4v+1}{4}$$

Cancel out 4, the greatest common factor in 4 and 4.

$$4v^{2}-15v-4=\left(v-4\right)\left(4v+1\right)$$

Evaluate

$$\left(v-4\right)\left(4v+1\right)$$