Solve 13c^2-7c-6

$$13 c ^ { 2 } - 7 c - 6$$

Factor

$$\left(c-1\right)\left(13c+6\right)$$

Solution Steps

Factor the expression by grouping. First, the expression needs to be rewritten as 13c^{2}+ac+bc-6. To find a and b, set up a system to be solved.

$$a+b=-7 ab=13\left(-6\right)=-78$$

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 -78.

$$1,-78 2,-39 3,-26 6,-13$$

Calculate the sum for each pair.

$$1-78=-77 2-39=-37 3-26=-23 6-13=-7$$

The solution is the pair that gives sum -7.

$$a=-13 b=6$$

Rewrite 13c^{2}-7c-6 as \left(13c^{2}-13c\right)+\left(6c-6\right).

$$\left(13c^{2}-13c\right)+\left(6c-6\right)$$

Factor out 13c in the first and 6 in the second group.

$$13c\left(c-1\right)+6\left(c-1\right)$$

Factor out common term c-1 by using distributive property.

$$\left(c-1\right)\left(13c+6\right)$$

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.

$$13c^{2}-7c-6=0$$

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.

$$c=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 13\left(-6\right)}}{2\times 13}$$

Square -7.

$$c=\frac{-\left(-7\right)±\sqrt{49-4\times 13\left(-6\right)}}{2\times 13}$$

Multiply -4 times 13.

$$c=\frac{-\left(-7\right)±\sqrt{49-52\left(-6\right)}}{2\times 13}$$

Multiply -52 times -6.

$$c=\frac{-\left(-7\right)±\sqrt{49+312}}{2\times 13}$$

Add 49 to 312.

$$c=\frac{-\left(-7\right)±\sqrt{361}}{2\times 13}$$

Take the square root of 361.

$$c=\frac{-\left(-7\right)±19}{2\times 13}$$

The opposite of -7 is 7.

$$c=\frac{7±19}{2\times 13}$$

Multiply 2 times 13.

$$c=\frac{7±19}{26}$$

Now solve the equation c=\frac{7±19}{26} when ± is plus. Add 7 to 19.

$$c=\frac{26}{26}$$

Divide 26 by 26.

$$c=1$$

Now solve the equation c=\frac{7±19}{26} when ± is minus. Subtract 19 from 7.

$$c=-\frac{12}{26}$$

Reduce the fraction \frac{-12}{26} to lowest terms by extracting and canceling out 2.

$$c=-\frac{6}{13}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 1 for x_{1} and -\frac{6}{13} for x_{2}.

$$13c^{2}-7c-6=13\left(c-1\right)\left(c-\left(-\frac{6}{13}\right)\right)$$

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

$$13c^{2}-7c-6=13\left(c-1\right)\left(c+\frac{6}{13}\right)$$

Add \frac{6}{13} to c by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$13c^{2}-7c-6=13\left(c-1\right)\times \frac{13c+6}{13}$$

Cancel out 13, the greatest common factor in 13 and 13.

$$13c^{2}-7c-6=\left(c-1\right)\left(13c+6\right)$$

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 13

$$x ^ 2 -\frac{7}{13}x -\frac{6}{13} = 0$$

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

$$r + s = \frac{7}{13} rs = -\frac{6}{13}$$

Two numbers r and s sum up to \frac{7}{13} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{13} = \frac{7}{26}. 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>

$$r = \frac{7}{26} - u s = \frac{7}{26} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{6}{13}

$$(\frac{7}{26} - u) (\frac{7}{26} + u) = -\frac{6}{13}$$

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

$$\frac{49}{676} - u^2 = -\frac{6}{13}$$

Simplify the expression by subtracting \frac{49}{676} on both sides

$$-u^2 = -\frac{6}{13}-\frac{49}{676} = -\frac{361}{676}$$

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

$$u^2 = \frac{361}{676} u = \pm\sqrt{\frac{361}{676}} = \pm \frac{19}{26}$$

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

$$r =\frac{7}{26} - \frac{19}{26} = -0.462 s = \frac{7}{26} + \frac{19}{26} = 1$$

Evaluate

$$\left(c-1\right)\left(13c+6\right)$$