Solve (-5y^2+7y-6)+(3y^2+4y+2)

$$( - 5 y ^ { 2 } + 7 y - 6 ) + ( 3 y ^ { 2 } + 4 y + 2 )$$

Evaluate

$$-2y^{2}+11y-4$$

Solution Steps

Combine -5y^{2} and 3y^{2} to get -2y^{2}.

$$-2y^{2}+7y-6+4y+2$$

Combine 7y and 4y to get 11y.

$$-2y^{2}+11y-6+2$$

Add -6 and 2 to get -4.

$$-2y^{2}+11y-4$$

Factor

$$-2\left(y-\frac{11-\sqrt{89}}{4}\right)\left(y-\frac{\sqrt{89}+11}{4}\right)$$

Steps Using the Quadratic Formula

Combine -5y^{2} and 3y^{2} to get -2y^{2}.

$$factor(-2y^{2}+7y-6+4y+2)$$

Combine 7y and 4y to get 11y.

$$factor(-2y^{2}+11y-6+2)$$

Add -6 and 2 to get -4.

$$factor(-2y^{2}+11y-4)$$

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.

$$-2y^{2}+11y-4=0$$

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.

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

Square 11.

$$y=\frac{-11±\sqrt{121-4\left(-2\right)\left(-4\right)}}{2\left(-2\right)}$$

Multiply -4 times -2.

$$y=\frac{-11±\sqrt{121+8\left(-4\right)}}{2\left(-2\right)}$$

Multiply 8 times -4.

$$y=\frac{-11±\sqrt{121-32}}{2\left(-2\right)}$$

Add 121 to -32.

$$y=\frac{-11±\sqrt{89}}{2\left(-2\right)}$$

Multiply 2 times -2.

$$y=\frac{-11±\sqrt{89}}{-4}$$

Now solve the equation y=\frac{-11±\sqrt{89}}{-4} when ± is plus. Add -11 to \sqrt{89}.

$$y=\frac{\sqrt{89}-11}{-4}$$

Divide -11+\sqrt{89} by -4.

$$y=\frac{11-\sqrt{89}}{4}$$

Now solve the equation y=\frac{-11±\sqrt{89}}{-4} when ± is minus. Subtract \sqrt{89} from -11.

$$y=\frac{-\sqrt{89}-11}{-4}$$

Divide -11-\sqrt{89} by -4.

$$y=\frac{\sqrt{89}+11}{4}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{11-\sqrt{89}}{4} for x_{1} and \frac{11+\sqrt{89}}{4} for x_{2}.

$$-2y^{2}+11y-4=-2\left(y-\frac{11-\sqrt{89}}{4}\right)\left(y-\frac{\sqrt{89}+11}{4}\right)$$