Solve 3y^2-10y-189

$$3 { y }^{ 2 } -10y-189$$

Factor

$$3\left(y-\frac{5-4\sqrt{37}}{3}\right)\left(y-\frac{4\sqrt{37}+5}{3}\right)$$

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.

$$3y^{2}-10y-189=0$$

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.

$$y=\frac{-\left(-10\right)±\sqrt{\left(-10\right)^{2}-4\times 3\left(-189\right)}}{2\times 3}$$

Square -10.

$$y=\frac{-\left(-10\right)±\sqrt{100-4\times 3\left(-189\right)}}{2\times 3}$$

Multiply -4 times 3.

$$y=\frac{-\left(-10\right)±\sqrt{100-12\left(-189\right)}}{2\times 3}$$

Multiply -12 times -189.

$$y=\frac{-\left(-10\right)±\sqrt{100+2268}}{2\times 3}$$

Add 100 to 2268.

$$y=\frac{-\left(-10\right)±\sqrt{2368}}{2\times 3}$$

Take the square root of 2368.

$$y=\frac{-\left(-10\right)±8\sqrt{37}}{2\times 3}$$

The opposite of -10 is 10.

$$y=\frac{10±8\sqrt{37}}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{10±8\sqrt{37}}{6}$$

Now solve the equation y=\frac{10±8\sqrt{37}}{6} when ± is plus. Add 10 to 8\sqrt{37}.

$$y=\frac{8\sqrt{37}+10}{6}$$

Divide 10+8\sqrt{37} by 6.

$$y=\frac{4\sqrt{37}+5}{3}$$

Now solve the equation y=\frac{10±8\sqrt{37}}{6} when ± is minus. Subtract 8\sqrt{37} from 10.

$$y=\frac{10-8\sqrt{37}}{6}$$

Divide 10-8\sqrt{37} by 6.

$$y=\frac{5-4\sqrt{37}}{3}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{5+4\sqrt{37}}{3} for x_{1} and \frac{5-4\sqrt{37}}{3} for x_{2}.

$$3y^{2}-10y-189=3\left(y-\frac{4\sqrt{37}+5}{3}\right)\left(y-\frac{5-4\sqrt{37}}{3}\right)$$

Evaluate

$$3y^{2}-10y-189$$