Solve 14n^2-82n-12

$$14 n ^ { 2 } - 82 n - 12$$

Factor

$$2\left(n-6\right)\left(7n+1\right)$$

Solution Steps

Factor out 2.

$$2\left(7n^{2}-41n-6\right)$$

Consider 7n^{2}-41n-6. Factor the expression by grouping. First, the expression needs to be rewritten as 7n^{2}+an+bn-6. To find a and b, set up a system to be solved.

$$a+b=-41 ab=7\left(-6\right)=-42$$

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 -42.

$$1,-42 2,-21 3,-14 6,-7$$

Calculate the sum for each pair.

$$1-42=-41 2-21=-19 3-14=-11 6-7=-1$$

The solution is the pair that gives sum -41.

$$a=-42 b=1$$

Rewrite 7n^{2}-41n-6 as \left(7n^{2}-42n\right)+\left(n-6\right).

$$\left(7n^{2}-42n\right)+\left(n-6\right)$$

Factor out 7n in 7n^{2}-42n.

$$7n\left(n-6\right)+n-6$$

Factor out common term n-6 by using distributive property.

$$\left(n-6\right)\left(7n+1\right)$$

Rewrite the complete factored expression.

$$2\left(n-6\right)\left(7n+1\right)$$

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.

$$14n^{2}-82n-12=0$$

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.

$$n=\frac{-\left(-82\right)±\sqrt{\left(-82\right)^{2}-4\times 14\left(-12\right)}}{2\times 14}$$

Square -82.

$$n=\frac{-\left(-82\right)±\sqrt{6724-4\times 14\left(-12\right)}}{2\times 14}$$

Multiply -4 times 14.

$$n=\frac{-\left(-82\right)±\sqrt{6724-56\left(-12\right)}}{2\times 14}$$

Multiply -56 times -12.

$$n=\frac{-\left(-82\right)±\sqrt{6724+672}}{2\times 14}$$

Add 6724 to 672.

$$n=\frac{-\left(-82\right)±\sqrt{7396}}{2\times 14}$$

Take the square root of 7396.

$$n=\frac{-\left(-82\right)±86}{2\times 14}$$

The opposite of -82 is 82.

$$n=\frac{82±86}{2\times 14}$$

Multiply 2 times 14.

$$n=\frac{82±86}{28}$$

Now solve the equation n=\frac{82±86}{28} when ± is plus. Add 82 to 86.

$$n=\frac{168}{28}$$

Divide 168 by 28.

$$n=6$$

Now solve the equation n=\frac{82±86}{28} when ± is minus. Subtract 86 from 82.

$$n=-\frac{4}{28}$$

Reduce the fraction \frac{-4}{28} to lowest terms by extracting and canceling out 4.

$$n=-\frac{1}{7}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 6 for x_{1} and -\frac{1}{7} for x_{2}.

$$14n^{2}-82n-12=14\left(n-6\right)\left(n-\left(-\frac{1}{7}\right)\right)$$

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

$$14n^{2}-82n-12=14\left(n-6\right)\left(n+\frac{1}{7}\right)$$

Add \frac{1}{7} to n by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$14n^{2}-82n-12=14\left(n-6\right)\times \frac{7n+1}{7}$$

Cancel out 7, the greatest common factor in 14 and 7.

$$14n^{2}-82n-12=2\left(n-6\right)\left(7n+1\right)$$

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 14

$$x ^ 2 -\frac{41}{7}x -\frac{6}{7} = 0$$

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

$$r + s = \frac{41}{7} rs = -\frac{6}{7}$$

Two numbers r and s sum up to \frac{41}{7} exactly when the average of the two numbers is \frac{1}{2}*\frac{41}{7} = \frac{41}{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>

$$r = \frac{41}{14} - u s = \frac{41}{14} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{6}{7}

$$(\frac{41}{14} - u) (\frac{41}{14} + u) = -\frac{6}{7}$$

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

$$\frac{1681}{196} - u^2 = -\frac{6}{7}$$

Simplify the expression by subtracting \frac{1681}{196} on both sides

$$-u^2 = -\frac{6}{7}-\frac{1681}{196} = -\frac{1849}{196}$$

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

$$u^2 = \frac{1849}{196} u = \pm\sqrt{\frac{1849}{196}} = \pm \frac{43}{14}$$

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

$$r =\frac{41}{14} - \frac{43}{14} = -0.143 s = \frac{41}{14} + \frac{43}{14} = 6$$

Evaluate

$$2\left(n-6\right)\left(7n+1\right)$$