Solve (6y^2+1)+(5y^2-7y-2)

$$( 6 y ^ { 2 } + 1 ) + ( 5 y ^ { 2 } - 7 y - 2 )$$

Evaluate

$$11y^{2}-7y-1$$

Solution Steps

Combine 6y^{2} and 5y^{2} to get 11y^{2}.

$$11y^{2}+1-7y-2$$

Subtract 2 from 1 to get -1.

$$11y^{2}-1-7y$$

Factor

$$11\left(y-\frac{7-\sqrt{93}}{22}\right)\left(y-\frac{\sqrt{93}+7}{22}\right)$$

Steps Using the Quadratic Formula

Combine 6y^{2} and 5y^{2} to get 11y^{2}.

$$factor(11y^{2}+1-7y-2)$$

Subtract 2 from 1 to get -1.

$$factor(11y^{2}-1-7y)$$

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.

$$11y^{2}-7y-1=0$$

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.

$$y=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\times 11\left(-1\right)}}{2\times 11}$$

Square -7.

$$y=\frac{-\left(-7\right)±\sqrt{49-4\times 11\left(-1\right)}}{2\times 11}$$

Multiply -4 times 11.

$$y=\frac{-\left(-7\right)±\sqrt{49-44\left(-1\right)}}{2\times 11}$$

Multiply -44 times -1.

$$y=\frac{-\left(-7\right)±\sqrt{49+44}}{2\times 11}$$

Add 49 to 44.

$$y=\frac{-\left(-7\right)±\sqrt{93}}{2\times 11}$$

The opposite of -7 is 7.

$$y=\frac{7±\sqrt{93}}{2\times 11}$$

Multiply 2 times 11.

$$y=\frac{7±\sqrt{93}}{22}$$

Now solve the equation y=\frac{7±\sqrt{93}}{22} when ± is plus. Add 7 to \sqrt{93}.

$$y=\frac{\sqrt{93}+7}{22}$$

Now solve the equation y=\frac{7±\sqrt{93}}{22} when ± is minus. Subtract \sqrt{93} from 7.

$$y=\frac{7-\sqrt{93}}{22}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{7+\sqrt{93}}{22} for x_{1} and \frac{7-\sqrt{93}}{22} for x_{2}.

$$11y^{2}-7y-1=11\left(y-\frac{\sqrt{93}+7}{22}\right)\left(y-\frac{7-\sqrt{93}}{22}\right)$$