Solve (-y^2-2y+3)+(-7y^2+4)

$$( - y ^ { 2 } - 2 y + 3 ) + ( - 7 y ^ { 2 } + 4 )$$

Evaluate

$$7-2y-8y^{2}$$

Solution Steps

Add 3 and 4 to get 7.

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

Combine -y^{2} and -7y^{2} to get -8y^{2}.

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

Factor

$$-8\left(y-\frac{-\sqrt{57}-1}{8}\right)\left(y-\frac{\sqrt{57}-1}{8}\right)$$

Steps Using the Quadratic Formula

Add 3 and 4 to get 7.

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

Combine -y^{2} and -7y^{2} to get -8y^{2}.

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

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.

$$-8y^{2}-2y+7=0$$

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.

$$y=\frac{-\left(-2\right)±\sqrt{\left(-2\right)^{2}-4\left(-8\right)\times 7}}{2\left(-8\right)}$$

Square -2.

$$y=\frac{-\left(-2\right)±\sqrt{4-4\left(-8\right)\times 7}}{2\left(-8\right)}$$

Multiply -4 times -8.

$$y=\frac{-\left(-2\right)±\sqrt{4+32\times 7}}{2\left(-8\right)}$$

Multiply 32 times 7.

$$y=\frac{-\left(-2\right)±\sqrt{4+224}}{2\left(-8\right)}$$

Add 4 to 224.

$$y=\frac{-\left(-2\right)±\sqrt{228}}{2\left(-8\right)}$$

Take the square root of 228.

$$y=\frac{-\left(-2\right)±2\sqrt{57}}{2\left(-8\right)}$$

The opposite of -2 is 2.

$$y=\frac{2±2\sqrt{57}}{2\left(-8\right)}$$

Multiply 2 times -8.

$$y=\frac{2±2\sqrt{57}}{-16}$$

Now solve the equation y=\frac{2±2\sqrt{57}}{-16} when ± is plus. Add 2 to 2\sqrt{57}.

$$y=\frac{2\sqrt{57}+2}{-16}$$

Divide 2+2\sqrt{57} by -16.

$$y=\frac{-\sqrt{57}-1}{8}$$

Now solve the equation y=\frac{2±2\sqrt{57}}{-16} when ± is minus. Subtract 2\sqrt{57} from 2.

$$y=\frac{2-2\sqrt{57}}{-16}$$

Divide 2-2\sqrt{57} by -16.

$$y=\frac{\sqrt{57}-1}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1-\sqrt{57}}{8} for x_{1} and \frac{-1+\sqrt{57}}{8} for x_{2}.

$$-8y^{2}-2y+7=-8\left(y-\frac{-\sqrt{57}-1}{8}\right)\left(y-\frac{\sqrt{57}-1}{8}\right)$$