Solve 3y^2-21y+33

$$3 y ^ { 2 } - 21 y + 33$$

Factor

$$3\left(y-\frac{7-\sqrt{5}}{2}\right)\left(y-\frac{\sqrt{5}+7}{2}\right)$$

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.

$$3y^{2}-21y+33=0$$

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.

$$y=\frac{-\left(-21\right)±\sqrt{\left(-21\right)^{2}-4\times 3\times 33}}{2\times 3}$$

Square -21.

$$y=\frac{-\left(-21\right)±\sqrt{441-4\times 3\times 33}}{2\times 3}$$

Multiply -4 times 3.

$$y=\frac{-\left(-21\right)±\sqrt{441-12\times 33}}{2\times 3}$$

Multiply -12 times 33.

$$y=\frac{-\left(-21\right)±\sqrt{441-396}}{2\times 3}$$

Add 441 to -396.

$$y=\frac{-\left(-21\right)±\sqrt{45}}{2\times 3}$$

Take the square root of 45.

$$y=\frac{-\left(-21\right)±3\sqrt{5}}{2\times 3}$$

The opposite of -21 is 21.

$$y=\frac{21±3\sqrt{5}}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{21±3\sqrt{5}}{6}$$

Now solve the equation y=\frac{21±3\sqrt{5}}{6} when ± is plus. Add 21 to 3\sqrt{5}.

$$y=\frac{3\sqrt{5}+21}{6}$$

Divide 21+3\sqrt{5} by 6.

$$y=\frac{\sqrt{5}+7}{2}$$

Now solve the equation y=\frac{21±3\sqrt{5}}{6} when ± is minus. Subtract 3\sqrt{5} from 21.

$$y=\frac{21-3\sqrt{5}}{6}$$

Divide 21-3\sqrt{5} by 6.

$$y=\frac{7-\sqrt{5}}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{5}}{2} for x_{1} and \frac{7-\sqrt{5}}{2} for x_{2}.

$$3y^{2}-21y+33=3\left(y-\frac{\sqrt{5}+7}{2}\right)\left(y-\frac{7-\sqrt{5}}{2}\right)$$

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 3

$$x ^ 2 -7x +11 = 0$$

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

$$r + s = 7 rs = 11$$

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

$$r = \frac{7}{2} - u s = \frac{7}{2} + u$$

To solve for unknown quantity u, substitute these in the product equation rs = 11

$$(\frac{7}{2} - u) (\frac{7}{2} + u) = 11$$

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

$$\frac{49}{4} - u^2 = 11$$

Simplify the expression by subtracting \frac{49}{4} on both sides

$$-u^2 = 11-\frac{49}{4} = -\frac{5}{4}$$

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

$$u^2 = \frac{5}{4} u = \pm\sqrt{\frac{5}{4}} = \pm \frac{\sqrt{5}}{2}$$

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

$$r =\frac{7}{2} - \frac{\sqrt{5}}{2} = 2.382 s = \frac{7}{2} + \frac{\sqrt{5}}{2} = 4.618$$

Evaluate

$$3\left(y^{2}-7y+11\right)$$