Solve 6y^2+8y-3y^2+7y-9

$$6 y ^ { 2 } + 8 y - 3 y ^ { 2 } + 7 y - 9$$

Evaluate

$$3\left(y^{2}+5y-3\right)$$

Solution Steps

Combine 6y^{2} and -3y^{2} to get 3y^{2}.

$$3y^{2}+8y+7y-9$$

Combine 8y and 7y to get 15y.

$$3y^{2}+15y-9$$

Factor

$$3\left(y-\frac{-\sqrt{37}-5}{2}\right)\left(y-\frac{\sqrt{37}-5}{2}\right)$$

Steps Using the Quadratic Formula

Combine 6y^{2} and -3y^{2} to get 3y^{2}.

$$factor(3y^{2}+8y+7y-9)$$

Combine 8y and 7y to get 15y.

$$factor(3y^{2}+15y-9)$$

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.

$$3y^{2}+15y-9=0$$

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.

$$y=\frac{-15±\sqrt{15^{2}-4\times 3\left(-9\right)}}{2\times 3}$$

Square 15.

$$y=\frac{-15±\sqrt{225-4\times 3\left(-9\right)}}{2\times 3}$$

Multiply -4 times 3.

$$y=\frac{-15±\sqrt{225-12\left(-9\right)}}{2\times 3}$$

Multiply -12 times -9.

$$y=\frac{-15±\sqrt{225+108}}{2\times 3}$$

Add 225 to 108.

$$y=\frac{-15±\sqrt{333}}{2\times 3}$$

Take the square root of 333.

$$y=\frac{-15±3\sqrt{37}}{2\times 3}$$

Multiply 2 times 3.

$$y=\frac{-15±3\sqrt{37}}{6}$$

Now solve the equation y=\frac{-15±3\sqrt{37}}{6} when ± is plus. Add -15 to 3\sqrt{37}.

$$y=\frac{3\sqrt{37}-15}{6}$$

Divide -15+3\sqrt{37} by 6.

$$y=\frac{\sqrt{37}-5}{2}$$

Now solve the equation y=\frac{-15±3\sqrt{37}}{6} when ± is minus. Subtract 3\sqrt{37} from -15.

$$y=\frac{-3\sqrt{37}-15}{6}$$

Divide -15-3\sqrt{37} by 6.

$$y=\frac{-\sqrt{37}-5}{2}$$

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

$$3y^{2}+15y-9=3\left(y-\frac{\sqrt{37}-5}{2}\right)\left(y-\frac{-\sqrt{37}-5}{2}\right)$$