Solve 6y^2-21y+12

$$6 y ^ { 2 } - 21 y + 12$$

Factor

$$6\left(y-\frac{7-\sqrt{17}}{4}\right)\left(y-\frac{\sqrt{17}+7}{4}\right)$$

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.

$$6y^{2}-21y+12=0$$

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.

$$y=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 6\times 12}}{2\times 6}$$

Square -21.

$$y=\frac{-\left(-21\right)±\sqrt{441-4\times 6\times 12}}{2\times 6}$$

Multiply -4 times 6.

$$y=\frac{-\left(-21\right)±\sqrt{441-24\times 12}}{2\times 6}$$

Multiply -24 times 12.

$$y=\frac{-\left(-21\right)±\sqrt{441-288}}{2\times 6}$$

Add 441 to -288.

$$y=\frac{-\left(-21\right)±\sqrt{153}}{2\times 6}$$

Take the square root of 153.

$$y=\frac{-\left(-21\right)±3\sqrt{17}}{2\times 6}$$

The opposite of -21 is 21.

$$y=\frac{21±3\sqrt{17}}{2\times 6}$$

Multiply 2 times 6.

$$y=\frac{21±3\sqrt{17}}{12}$$

Now solve the equation y=\frac{21±3\sqrt{17}}{12} when ± is plus. Add 21 to 3\sqrt{17}.

$$y=\frac{3\sqrt{17}+21}{12}$$

Divide 21+3\sqrt{17} by 12.

$$y=\frac{\sqrt{17}+7}{4}$$

Now solve the equation y=\frac{21±3\sqrt{17}}{12} when ± is minus. Subtract 3\sqrt{17} from 21.

$$y=\frac{21-3\sqrt{17}}{12}$$

Divide 21-3\sqrt{17} by 12.

$$y=\frac{7-\sqrt{17}}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{17}}{4} for x_{1} and \frac{7-\sqrt{17}}{4} for x_{2}.

$$6y^{2}-21y+12=6\left(y-\frac{\sqrt{17}+7}{4}\right)\left(y-\frac{7-\sqrt{17}}{4}\right)$$

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

$$x ^ 2 -\frac{7}{2}x +2 = 0$$

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

$$r + s = \frac{7}{2} rs = 2$$

Two numbers r and s sum up to \frac{7}{2} exactly when the average of the two numbers is \frac{1}{2}*\frac{7}{2} = \frac{7}{4}. 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>

$$r = \frac{7}{4} - u s = \frac{7}{4} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 2

$$(\frac{7}{4} - u) (\frac{7}{4} + u) = 2$$

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

$$\frac{49}{16} - u^2 = 2$$

Simplify the expression by subtracting \frac{49}{16} on both sides

$$-u^2 = 2-\frac{49}{16} = -\frac{17}{16}$$

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

$$u^2 = \frac{17}{16} u = \pm\sqrt{\frac{17}{16}} = \pm \frac{\sqrt{17}}{4}$$

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

$$r =\frac{7}{4} - \frac{\sqrt{17}}{4} = 0.719 s = \frac{7}{4} + \frac{\sqrt{17}}{4} = 2.781$$

Evaluate

$$6y^{2}-21y+12$$