Solve y^2+y-4+y+4y^2

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

Evaluate

$$5y^{2}+2y-4$$

Solution Steps

Combine y and y to get 2y.

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

Combine y^{2} and 4y^{2} to get 5y^{2}.

$$5y^{2}+2y-4$$

Factor

$$5\left(y-\frac{-\sqrt{21}-1}{5}\right)\left(y-\frac{\sqrt{21}-1}{5}\right)$$

Steps Using the Quadratic Formula

Combine y and y to get 2y.

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

Combine y^{2} and 4y^{2} to get 5y^{2}.

$$factor(5y^{2}+2y-4)$$

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.

$$5y^{2}+2y-4=0$$

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.

$$y=\frac{-2±\sqrt{2^{2}-4\times 5\left(-4\right)}}{2\times 5}$$

Square 2.

$$y=\frac{-2±\sqrt{4-4\times 5\left(-4\right)}}{2\times 5}$$

Multiply -4 times 5.

$$y=\frac{-2±\sqrt{4-20\left(-4\right)}}{2\times 5}$$

Multiply -20 times -4.

$$y=\frac{-2±\sqrt{4+80}}{2\times 5}$$

Add 4 to 80.

$$y=\frac{-2±\sqrt{84}}{2\times 5}$$

Take the square root of 84.

$$y=\frac{-2±2\sqrt{21}}{2\times 5}$$

Multiply 2 times 5.

$$y=\frac{-2±2\sqrt{21}}{10}$$

Now solve the equation y=\frac{-2±2\sqrt{21}}{10} when ± is plus. Add -2 to 2\sqrt{21}.

$$y=\frac{2\sqrt{21}-2}{10}$$

Divide -2+2\sqrt{21} by 10.

$$y=\frac{\sqrt{21}-1}{5}$$

Now solve the equation y=\frac{-2±2\sqrt{21}}{10} when ± is minus. Subtract 2\sqrt{21} from -2.

$$y=\frac{-2\sqrt{21}-2}{10}$$

Divide -2-2\sqrt{21} by 10.

$$y=\frac{-\sqrt{21}-1}{5}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute \frac{-1+\sqrt{21}}{5} for x_{1} and \frac{-1-\sqrt{21}}{5} for x_{2}.

$$5y^{2}+2y-4=5\left(y-\frac{\sqrt{21}-1}{5}\right)\left(y-\frac{-\sqrt{21}-1}{5}\right)$$