Solve 12x^2+92x-433=

$$12 x ^ { 2 } + 92 x - 433 =$$

Factor

$$12\left(x-\left(-\frac{\sqrt{457}}{3}-\frac{23}{6}\right)\right)\left(x-\left(\frac{\sqrt{457}}{3}-\frac{23}{6}\right)\right)$$

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.

$$12x^{2}+92x-433=0$$

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.

$$x=\frac{-92±\sqrt{92^{2}-4\times 12\left(-433\right)}}{2\times 12}$$

Square 92.

$$x=\frac{-92±\sqrt{8464-4\times 12\left(-433\right)}}{2\times 12}$$

Multiply -4 times 12.

$$x=\frac{-92±\sqrt{8464-48\left(-433\right)}}{2\times 12}$$

Multiply -48 times -433.

$$x=\frac{-92±\sqrt{8464+20784}}{2\times 12}$$

Add 8464 to 20784.

$$x=\frac{-92±\sqrt{29248}}{2\times 12}$$

Take the square root of 29248.

$$x=\frac{-92±8\sqrt{457}}{2\times 12}$$

Multiply 2 times 12.

$$x=\frac{-92±8\sqrt{457}}{24}$$

Now solve the equation x=\frac{-92±8\sqrt{457}}{24} when ± is plus. Add -92 to 8\sqrt{457}.

$$x=\frac{8\sqrt{457}-92}{24}$$

Divide -92+8\sqrt{457} by 24.

$$x=\frac{\sqrt{457}}{3}-\frac{23}{6}$$

Now solve the equation x=\frac{-92±8\sqrt{457}}{24} when ± is minus. Subtract 8\sqrt{457} from -92.

$$x=\frac{-8\sqrt{457}-92}{24}$$

Divide -92-8\sqrt{457} by 24.

$$x=-\frac{\sqrt{457}}{3}-\frac{23}{6}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute -\frac{23}{6}+\frac{\sqrt{457}}{3} for x_{1} and -\frac{23}{6}-\frac{\sqrt{457}}{3} for x_{2}.

$$12x^{2}+92x-433=12\left(x-\left(\frac{\sqrt{457}}{3}-\frac{23}{6}\right)\right)\left(x-\left(-\frac{\sqrt{457}}{3}-\frac{23}{6}\right)\right)$$

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 12

$$x ^ 2 +\frac{23}{3}x -\frac{433}{12} = 0$$

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

$$r + s = -\frac{23}{3} rs = -\frac{433}{12}$$

Two numbers r and s sum up to -\frac{23}{3} exactly when the average of the two numbers is \frac{1}{2}*-\frac{23}{3} = -\frac{23}{6}. 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>

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

To solve for unknown quantity u, substitute these in the product equation rs = -\frac{433}{12}

$$(-\frac{23}{6} - u) (-\frac{23}{6} + u) = -\frac{433}{12}$$

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

$$\frac{529}{36} - u^2 = -\frac{433}{12}$$

Simplify the expression by subtracting \frac{529}{36} on both sides

$$-u^2 = -\frac{433}{12}-\frac{529}{36} = -\frac{457}{9}$$

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

$$u^2 = \frac{457}{9} u = \pm\sqrt{\frac{457}{9}} = \pm \frac{\sqrt{457}}{3}$$

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

$$r =-\frac{23}{6} - \frac{\sqrt{457}}{3} = -10.959 s = -\frac{23}{6} + \frac{\sqrt{457}}{3} = 3.293$$

Evaluate

$$12x^{2}+92x-433$$