Solve 12k^2-106k-156=0

$$12 k ^ { 2 } - 106 k - 156 = 0$$

Solve for k

$$k = \frac{\sqrt{4681} + 53}{12} \approx 10.118152828$$

$$k=\frac{53-\sqrt{4681}}{12}\approx -1.284819494$$

Steps Using the Quadratic Formula

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.

$$12k^{2}-106k-156=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 12 for a, -106 for b, and -156 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$k=\frac{-\left(-106\right)±\sqrt{\left(-106\right)^{2}-4\times 12\left(-156\right)}}{2\times 12}$$

Square -106.

$$k=\frac{-\left(-106\right)±\sqrt{11236-4\times 12\left(-156\right)}}{2\times 12}$$

Multiply -4 times 12.

$$k=\frac{-\left(-106\right)±\sqrt{11236-48\left(-156\right)}}{2\times 12}$$

Multiply -48 times -156.

$$k=\frac{-\left(-106\right)±\sqrt{11236+7488}}{2\times 12}$$

Add 11236 to 7488.

$$k=\frac{-\left(-106\right)±\sqrt{18724}}{2\times 12}$$

Take the square root of 18724.

$$k=\frac{-\left(-106\right)±2\sqrt{4681}}{2\times 12}$$

The opposite of -106 is 106.

$$k=\frac{106±2\sqrt{4681}}{2\times 12}$$

Multiply 2 times 12.

$$k=\frac{106±2\sqrt{4681}}{24}$$

Now solve the equation k=\frac{106±2\sqrt{4681}}{24} when ± is plus. Add 106 to 2\sqrt{4681}.

$$k=\frac{2\sqrt{4681}+106}{24}$$

Divide 106+2\sqrt{4681} by 24.

$$k=\frac{\sqrt{4681}+53}{12}$$

Now solve the equation k=\frac{106±2\sqrt{4681}}{24} when ± is minus. Subtract 2\sqrt{4681} from 106.

$$k=\frac{106-2\sqrt{4681}}{24}$$

Divide 106-2\sqrt{4681} by 24.

$$k=\frac{53-\sqrt{4681}}{12}$$

The equation is now solved.

$$k=\frac{\sqrt{4681}+53}{12} k=\frac{53-\sqrt{4681}}{12}$$

Steps for Completing the Square

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$12k^{2}-106k-156=0$$

Add 156 to both sides of the equation.

$$12k^{2}-106k-156-\left(-156\right)=-\left(-156\right)$$

Subtracting -156 from itself leaves 0.

$$12k^{2}-106k=-\left(-156\right)$$

Subtract -156 from 0.

$$12k^{2}-106k=156$$

Divide both sides by 12.

$$\frac{12k^{2}-106k}{12}=\frac{156}{12}$$

Dividing by 12 undoes the multiplication by 12.

$$k^{2}+\left(-\frac{106}{12}\right)k=\frac{156}{12}$$

Reduce the fraction \frac{-106}{12} to lowest terms by extracting and canceling out 2.

$$k^{2}-\frac{53}{6}k=\frac{156}{12}$$

Divide 156 by 12.

$$k^{2}-\frac{53}{6}k=13$$

Divide -\frac{53}{6}, the coefficient of the x term, by 2 to get -\frac{53}{12}. Then add the square of -\frac{53}{12} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$k^{2}-\frac{53}{6}k+\left(-\frac{53}{12}\right)^{2}=13+\left(-\frac{53}{12}\right)^{2}$$

Square -\frac{53}{12} by squaring both the numerator and the denominator of the fraction.

$$k^{2}-\frac{53}{6}k+\frac{2809}{144}=13+\frac{2809}{144}$$

Add 13 to \frac{2809}{144}.

$$k^{2}-\frac{53}{6}k+\frac{2809}{144}=\frac{4681}{144}$$

Factor k^{2}-\frac{53}{6}k+\frac{2809}{144}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(k-\frac{53}{12}\right)^{2}=\frac{4681}{144}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(k-\frac{53}{12}\right)^{2}}=\sqrt{\frac{4681}{144}}$$

Simplify.

$$k-\frac{53}{12}=\frac{\sqrt{4681}}{12} k-\frac{53}{12}=-\frac{\sqrt{4681}}{12}$$

Add \frac{53}{12} to both sides of the equation.

$$k=\frac{\sqrt{4681}+53}{12} k=\frac{53-\sqrt{4681}}{12}$$

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 12

$$x ^ 2 -\frac{53}{6}x -13 = 0$$

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

$$r + s = \frac{53}{6} rs = -13$$

Two numbers r and s sum up to \frac{53}{6} exactly when the average of the two numbers is \frac{1}{2}*\frac{53}{6} = \frac{53}{12}. 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-gzdabgg4ehffg0hf.b01.azurefd.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{53}{12} - u s = \frac{53}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -13

$$(\frac{53}{12} - u) (\frac{53}{12} + u) = -13$$

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

$$\frac{2809}{144} - u^2 = -13$$

Simplify the expression by subtracting \frac{2809}{144} on both sides

$$-u^2 = -13-\frac{2809}{144} = -\frac{4681}{144}$$

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

$$u^2 = \frac{4681}{144} u = \pm\sqrt{\frac{4681}{144}} = \pm \frac{\sqrt{4681}}{12}$$

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

$$r =\frac{53}{12} - \frac{\sqrt{4681}}{12} = -1.285 s = \frac{53}{12} + \frac{\sqrt{4681}}{12} = 10.118$$