Solve 6x^2-23x-6

$$6 x ^ { 2 } - 23 x - 6$$

Factor

$$6\left(x-\frac{23-\sqrt{673}}{12}\right)\left(x-\frac{\sqrt{673}+23}{12}\right)$$

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.

$$6x^{2}-23x-6=0$$

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.

$$x=\frac{-\left(-23\right)±\sqrt{\left(-23\right)^{2}-4\times 6\left(-6\right)}}{2\times 6}$$

Square -23.

$$x=\frac{-\left(-23\right)±\sqrt{529-4\times 6\left(-6\right)}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-23\right)±\sqrt{529-24\left(-6\right)}}{2\times 6}$$

Multiply -24 times -6.

$$x=\frac{-\left(-23\right)±\sqrt{529+144}}{2\times 6}$$

Add 529 to 144.

$$x=\frac{-\left(-23\right)±\sqrt{673}}{2\times 6}$$

The opposite of -23 is 23.

$$x=\frac{23±\sqrt{673}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{23±\sqrt{673}}{12}$$

Now solve the equation x=\frac{23±\sqrt{673}}{12} when ± is plus. Add 23 to \sqrt{673}.

$$x=\frac{\sqrt{673}+23}{12}$$

Now solve the equation x=\frac{23±\sqrt{673}}{12} when ± is minus. Subtract \sqrt{673} from 23.

$$x=\frac{23-\sqrt{673}}{12}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{23+\sqrt{673}}{12} for x_{1} and \frac{23-\sqrt{673}}{12} for x_{2}.

$$6x^{2}-23x-6=6\left(x-\frac{\sqrt{673}+23}{12}\right)\left(x-\frac{23-\sqrt{673}}{12}\right)$$

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 6

$$x ^ 2 -\frac{23}{6}x -1 = 0$$

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

$$r + s = \frac{23}{6} rs = -1$$

Two numbers r and s sum up to \frac{23}{6} exactly when the average of the two numbers is \frac{1}{2}*\frac{23}{6} = \frac{23}{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.azureedge.net/customsolver/quadraticgraph.png' style='width: 100%;max-width: 700px' /></div>

$$r = \frac{23}{12} - u s = \frac{23}{12} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = -1

$$(\frac{23}{12} - u) (\frac{23}{12} + u) = -1$$

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

$$\frac{529}{144} - u^2 = -1$$

Simplify the expression by subtracting \frac{529}{144} on both sides

$$-u^2 = -1-\frac{529}{144} = -\frac{673}{144}$$

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

$$u^2 = \frac{673}{144} u = \pm\sqrt{\frac{673}{144}} = \pm \frac{\sqrt{673}}{12}$$

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

$$r =\frac{23}{12} - \frac{\sqrt{673}}{12} = -0.245 s = \frac{23}{12} + \frac{\sqrt{673}}{12} = 4.079$$

Evaluate

$$6x^{2}-23x-6$$