Solve 6x^2-24x+10

$$6 x ^ { 2 } - 24 x + 10$$

Factor

$$6\left(x-\left(-\frac{\sqrt{21}}{3}+2\right)\right)\left(x-\left(\frac{\sqrt{21}}{3}+2\right)\right)$$

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.

$$6x^{2}-24x+10=0$$

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.

$$x=\frac{-\left(-24\right)±\sqrt{\left(-24\right)^{2}-4\times 6\times 10}}{2\times 6}$$

Square -24.

$$x=\frac{-\left(-24\right)±\sqrt{576-4\times 6\times 10}}{2\times 6}$$

Multiply -4 times 6.

$$x=\frac{-\left(-24\right)±\sqrt{576-24\times 10}}{2\times 6}$$

Multiply -24 times 10.

$$x=\frac{-\left(-24\right)±\sqrt{576-240}}{2\times 6}$$

Add 576 to -240.

$$x=\frac{-\left(-24\right)±\sqrt{336}}{2\times 6}$$

Take the square root of 336.

$$x=\frac{-\left(-24\right)±4\sqrt{21}}{2\times 6}$$

The opposite of -24 is 24.

$$x=\frac{24±4\sqrt{21}}{2\times 6}$$

Multiply 2 times 6.

$$x=\frac{24±4\sqrt{21}}{12}$$

Now solve the equation x=\frac{24±4\sqrt{21}}{12} when ± is plus. Add 24 to 4\sqrt{21}.

$$x=\frac{4\sqrt{21}+24}{12}$$

Divide 24+4\sqrt{21} by 12.

$$x=\frac{\sqrt{21}}{3}+2$$

Now solve the equation x=\frac{24±4\sqrt{21}}{12} when ± is minus. Subtract 4\sqrt{21} from 24.

$$x=\frac{24-4\sqrt{21}}{12}$$

Divide 24-4\sqrt{21} by 12.

$$x=-\frac{\sqrt{21}}{3}+2$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 2+\frac{\sqrt{21}}{3} for x_{1} and 2-\frac{\sqrt{21}}{3} for x_{2}.

$$6x^{2}-24x+10=6\left(x-\left(\frac{\sqrt{21}}{3}+2\right)\right)\left(x-\left(-\frac{\sqrt{21}}{3}+2\right)\right)$$

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 6

$$x ^ 2 -4x +\frac{5}{3} = 0$$

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

$$r + s = 4 rs = \frac{5}{3}$$

Two numbers r and s sum up to 4 exactly when the average of the two numbers is \frac{1}{2}*4 = 2. 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>

$$r = 2 - u s = 2 + u$$

To solve for unknown quantity u, substitute these in the product equation rs = \frac{5}{3}

$$(2 - u) (2 + u) = \frac{5}{3}$$

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

$$4 - u^2 = \frac{5}{3}$$

Simplify the expression by subtracting 4 on both sides

$$-u^2 = \frac{5}{3}-4 = -\frac{7}{3}$$

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

$$u^2 = \frac{7}{3} u = \pm\sqrt{\frac{7}{3}} = \pm \frac{\sqrt{7}}{\sqrt{3}}$$

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

$$r =2 - \frac{\sqrt{7}}{\sqrt{3}} = 0.472 s = 2 + \frac{\sqrt{7}}{\sqrt{3}} = 3.528$$

Evaluate

$$6x^{2}-24x+10$$