Solve 3y^2-10y-24div3y-4

$$3 { y }^{ 2 } -10y-24 \div 3y-4$$

Evaluate

$$3y^{2}-18y-4$$

Solution Steps

Divide 24 by 3 to get 8.

$$3y^{2}-10y-8y-4$$

Combine -10y and -8y to get -18y.

$$3y^{2}-18y-4$$

Factor

$$3\left(y-\left(-\frac{\sqrt{93}}{3}+3\right)\right)\left(y-\left(\frac{\sqrt{93}}{3}+3\right)\right)$$

Steps Using the Quadratic Formula

Divide 24 by 3 to get 8.

$$factor(3y^{2}-10y-8y-4)$$

Combine -10y and -8y to get -18y.

$$factor(3y^{2}-18y-4)$$

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.

$$3y^{2}-18y-4=0$$

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.

$$y=\frac{-\left(-18\right)±\sqrt{\left(-18\right)^{2}-4\times 3\left(-4\right)}}{2\times 3}$$

Square -18.

$$y=\frac{-\left(-18\right)±\sqrt{324-4\times 3\left(-4\right)}}{2\times 3}$$

Multiply -4 times 3.

$$y=\frac{-\left(-18\right)±\sqrt{324-12\left(-4\right)}}{2\times 3}$$

Multiply -12 times -4.

$$y=\frac{-\left(-18\right)±\sqrt{324+48}}{2\times 3}$$

Add 324 to 48.

$$y=\frac{-\left(-18\right)±\sqrt{372}}{2\times 3}$$

Take the square root of 372.

$$y=\frac{-\left(-18\right)±2\sqrt{93}}{2\times 3}$$

The opposite of -18 is 18.

$$y=\frac{18±2\sqrt{93}}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{18±2\sqrt{93}}{6}$$

Now solve the equation y=\frac{18±2\sqrt{93}}{6} when ± is plus. Add 18 to 2\sqrt{93}.

$$y=\frac{2\sqrt{93}+18}{6}$$

Divide 18+2\sqrt{93} by 6.

$$y=\frac{\sqrt{93}}{3}+3$$

Now solve the equation y=\frac{18±2\sqrt{93}}{6} when ± is minus. Subtract 2\sqrt{93} from 18.

$$y=\frac{18-2\sqrt{93}}{6}$$

Divide 18-2\sqrt{93} by 6.

$$y=-\frac{\sqrt{93}}{3}+3$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 3+\frac{\sqrt{93}}{3} for x_{1} and 3-\frac{\sqrt{93}}{3} for x_{2}.

$$3y^{2}-18y-4=3\left(y-\left(\frac{\sqrt{93}}{3}+3\right)\right)\left(y-\left(-\frac{\sqrt{93}}{3}+3\right)\right)$$