Solve 81y^2+31y

$$81 y ^ { 2 } + 31 y$$

Factor

$$y\left(81y+31\right)$$

Solution Steps

Factor out y.

$$y\left(81y+31\right)$$

Steps Using the Quadratic Formula

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.

$$81y^{2}+31y=0$$

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.

$$y=\frac{-31±\sqrt{31^{2}}}{2\times 81}$$

Take the square root of 31^{2}.

$$y=\frac{-31±31}{2\times 81}$$

Multiply 2 times 81.

$$y=\frac{-31±31}{162}$$

Now solve the equation y=\frac{-31±31}{162} when ± is plus. Add -31 to 31.

$$y=\frac{0}{162}$$

Divide 0 by 162.

$$y=0$$

Now solve the equation y=\frac{-31±31}{162} when ± is minus. Subtract 31 from -31.

$$y=-\frac{62}{162}$$

Reduce the fraction \frac{-62}{162} to lowest terms by extracting and canceling out 2.

$$y=-\frac{31}{81}$$

Factor the original expression using ax^{2}+bx+c=a\left(x-x_{1}\right)\left(x-x_{2}\right). Substitute 0 for x_{1} and -\frac{31}{81} for x_{2}.

$$81y^{2}+31y=81y\left(y-\left(-\frac{31}{81}\right)\right)$$

Simplify all the expressions of the form p-\left(-q\right) to p+q.

$$81y^{2}+31y=81y\left(y+\frac{31}{81}\right)$$

Add \frac{31}{81} to y by finding a common denominator and adding the numerators. Then reduce the fraction to lowest terms if possible.

$$81y^{2}+31y=81y\times \frac{81y+31}{81}$$

Cancel out 81, the greatest common factor in 81 and 81.

$$81y^{2}+31y=y\left(81y+31\right)$$

Evaluate

$$y\left(81y+31\right)$$