Solve x^2-24x+6

$$x ^ { 2 } - 24 x + 6$$

Factor

$$\left(x-\left(12-\sqrt{138}\right)\right)\left(x-\left(\sqrt{138}+12\right)\right)$$

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.

$$x^{2}-24x+6=0$$

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.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 6}}{2}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 6}}{2}$$

Multiply -4 times 6.

$$x=\frac{-\left(-24\right)±\sqrt{576-24}}{2}$$

Add 576 to -24.

$$x=\frac{-\left(-24\right)±\sqrt{552}}{2}$$

Take the square root of 552.

$$x=\frac{-\left(-24\right)±2\sqrt{138}}{2}$$

The opposite of -24 is 24.

$$x=\frac{24±2\sqrt{138}}{2}$$

Now solve the equation x=\frac{24±2\sqrt{138}}{2} when ± is plus. Add 24 to 2\sqrt{138}.

$$x=\frac{2\sqrt{138}+24}{2}$$

Divide 24+2\sqrt{138} by 2.

$$x=\sqrt{138}+12$$

Now solve the equation x=\frac{24±2\sqrt{138}}{2} when ± is minus. Subtract 2\sqrt{138} from 24.

$$x=\frac{24-2\sqrt{138}}{2}$$

Divide 24-2\sqrt{138} by 2.

$$x=12-\sqrt{138}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 12+\sqrt{138} for x_{1} and 12-\sqrt{138} for x_{2}.

$$x^{2}-24x+6=\left(x-\left(\sqrt{138}+12\right)\right)\left(x-\left(12-\sqrt{138}\right)\right)$$

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.

$$x ^ 2 -24x +6 = 0$$

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

$$r + s = 24 rs = 6$$

Two numbers r and s sum up to 24 exactly when the average of the two numbers is \frac{1}{2}*24 = 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>

$$r = 12 - u s = 12 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 6

$$(12 - u) (12 + u) = 6$$

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

$$144 - u^2 = 6$$

Simplify the expression by subtracting 144 on both sides

$$-u^2 = 6-144 = -138$$

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

$$u^2 = 138 u = \pm\sqrt{138} = \pm \sqrt{138}$$

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

$$r =12 - \sqrt{138} = 0.253 s = 12 + \sqrt{138} = 23.747$$

Evaluate

$$x^{2}-24x+6$$