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

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

Evaluate

$$3-7y-y^{2}$$

Solution Steps

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

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

Combine -3y and -4y to get -7y.

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

Factor

$$-\left(y-\frac{-\sqrt{61}-7}{2}\right)\left(y-\frac{\sqrt{61}-7}{2}\right)$$

Steps Using the Quadratic Formula

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

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

Combine -3y and -4y to get -7y.

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

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.

$$-y^{2}-7y+3=0$$

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.

$$y=\frac{-\left(-7\right)±\sqrt{\left(-7\right)^{2}-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Square -7.

$$y=\frac{-\left(-7\right)±\sqrt{49-4\left(-1\right)\times 3}}{2\left(-1\right)}$$

Multiply -4 times -1.

$$y=\frac{-\left(-7\right)±\sqrt{49+4\times 3}}{2\left(-1\right)}$$

Multiply 4 times 3.

$$y=\frac{-\left(-7\right)±\sqrt{49+12}}{2\left(-1\right)}$$

Add 49 to 12.

$$y=\frac{-\left(-7\right)±\sqrt{61}}{2\left(-1\right)}$$

The opposite of -7 is 7.

$$y=\frac{7±\sqrt{61}}{2\left(-1\right)}$$

Multiply 2 times -1.

$$y=\frac{7±\sqrt{61}}{-2}$$

Now solve the equation y=\frac{7±\sqrt{61}}{-2} when ± is plus. Add 7 to \sqrt{61}.

$$y=\frac{\sqrt{61}+7}{-2}$$

Divide 7+\sqrt{61} by -2.

$$y=\frac{-\sqrt{61}-7}{2}$$

Now solve the equation y=\frac{7±\sqrt{61}}{-2} when ± is minus. Subtract \sqrt{61} from 7.

$$y=\frac{7-\sqrt{61}}{-2}$$

Divide 7-\sqrt{61} by -2.

$$y=\frac{\sqrt{61}-7}{2}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-7-\sqrt{61}}{2} for x_{1} and \frac{-7+\sqrt{61}}{2} for x_{2}.

$$-y^{2}-7y+3=-\left(y-\frac{-\sqrt{61}-7}{2}\right)\left(y-\frac{\sqrt{61}-7}{2}\right)$$