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

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

Evaluate

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

Solution Steps

Combine -6y and 3y to get -3y.

$$-3y+6-4y^{2}-2$$

Subtract 2 from 6 to get 4.

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

Factor

$$-4\left(y-\frac{-\sqrt{73}-3}{8}\right)\left(y-\frac{\sqrt{73}-3}{8}\right)$$

Steps Using the Quadratic Formula

Combine -6y and 3y to get -3y.

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

Subtract 2 from 6 to get 4.

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

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.

$$-4y^{2}-3y+4=0$$

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.

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

Square -3.

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

Multiply -4 times -4.

$$y=\frac{-\left(-3\right)±\sqrt{9+16\times 4}}{2\left(-4\right)}$$

Multiply 16 times 4.

$$y=\frac{-\left(-3\right)±\sqrt{9+64}}{2\left(-4\right)}$$

Add 9 to 64.

$$y=\frac{-\left(-3\right)±\sqrt{73}}{2\left(-4\right)}$$

The opposite of -3 is 3.

$$y=\frac{3±\sqrt{73}}{2\left(-4\right)}$$

Multiply 2 times -4.

$$y=\frac{3±\sqrt{73}}{-8}$$

Now solve the equation y=\frac{3±\sqrt{73}}{-8} when ± is plus. Add 3 to \sqrt{73}.

$$y=\frac{\sqrt{73}+3}{-8}$$

Divide 3+\sqrt{73} by -8.

$$y=\frac{-\sqrt{73}-3}{8}$$

Now solve the equation y=\frac{3±\sqrt{73}}{-8} when ± is minus. Subtract \sqrt{73} from 3.

$$y=\frac{3-\sqrt{73}}{-8}$$

Divide 3-\sqrt{73} by -8.

$$y=\frac{\sqrt{73}-3}{8}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-3-\sqrt{73}}{8} for x_{1} and \frac{-3+\sqrt{73}}{8} for x_{2}.

$$-4y^{2}-3y+4=-4\left(y-\frac{-\sqrt{73}-3}{8}\right)\left(y-\frac{\sqrt{73}-3}{8}\right)$$