Solve -by+b=-b(y+b)

$$- b y + b = - b ( y + b )$$

Solve for b

$$b=-1$$

$$b=0$$

Steps Using Factoring

Use the distributive property to multiply -b by y+b.

$$\left(-b\right)y+b=\left(-b\right)y+\left(-b\right)b$$

Subtract \left(-b\right)y from both sides.

$$\left(-b\right)y+b-\left(-b\right)y=\left(-b\right)b$$

Combine \left(-b\right)y and -\left(-b\right)y to get 0.

$$b=\left(-b\right)b$$

Subtract \left(-b\right)b from both sides.

$$b-\left(-b\right)b=0$$

Multiply b and b to get b^{2}.

$$b-\left(-b^{2}\right)=0$$

Multiply -1 and -1 to get 1.

$$b+b^{2}=0$$

Factor out b.

$$b\left(1+b\right)=0$$

To find equation solutions, solve b=0 and 1+b=0.

$$b=0 b=-1$$

Steps Using the Quadratic Formula

Use the distributive property to multiply -b by y+b.

$$\left(-b\right)y+b=\left(-b\right)y+\left(-b\right)b$$

Subtract \left(-b\right)y from both sides.

$$\left(-b\right)y+b-\left(-b\right)y=\left(-b\right)b$$

Combine \left(-b\right)y and -\left(-b\right)y to get 0.

$$b=\left(-b\right)b$$

Subtract \left(-b\right)b from both sides.

$$b-\left(-b\right)b=0$$

Multiply b and b to get b^{2}.

$$b-\left(-b^{2}\right)=0$$

Multiply -1 and -1 to get 1.

$$b+b^{2}=0$$

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.

$$b^{2}+b=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, 1 for b, and 0 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$b=\frac{-1±\sqrt{1^{2}}}{2}$$

Take the square root of 1^{2}.

$$b=\frac{-1±1}{2}$$

Now solve the equation b=\frac{-1±1}{2} when ± is plus. Add -1 to 1.

$$b=\frac{0}{2}$$

Divide 0 by 2.

$$b=0$$

Now solve the equation b=\frac{-1±1}{2} when ± is minus. Subtract 1 from -1.

$$b=-\frac{2}{2}$$

Divide -2 by 2.

$$b=-1$$

The equation is now solved.

$$b=0 b=-1$$

Steps for Completing the Square

Use the distributive property to multiply -b by y+b.

$$\left(-b\right)y+b=\left(-b\right)y+\left(-b\right)b$$

Subtract \left(-b\right)y from both sides.

$$\left(-b\right)y+b-\left(-b\right)y=\left(-b\right)b$$

Combine \left(-b\right)y and -\left(-b\right)y to get 0.

$$b=\left(-b\right)b$$

Subtract \left(-b\right)b from both sides.

$$b-\left(-b\right)b=0$$

Multiply b and b to get b^{2}.

$$b-\left(-b^{2}\right)=0$$

Multiply -1 and -1 to get 1.

$$b+b^{2}=0$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$b^{2}+b=0$$

Divide 1, the coefficient of the x term, by 2 to get \frac{1}{2}. Then add the square of \frac{1}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$b^{2}+b+\left(\frac{1}{2}\right)^{2}=\left(\frac{1}{2}\right)^{2}$$

Square \frac{1}{2} by squaring both the numerator and the denominator of the fraction.

$$b^{2}+b+\frac{1}{4}=\frac{1}{4}$$

Factor b^{2}+b+\frac{1}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(b+\frac{1}{2}\right)^{2}=\frac{1}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(b+\frac{1}{2}\right)^{2}}=\sqrt{\frac{1}{4}}$$

Simplify.

$$b+\frac{1}{2}=\frac{1}{2} b+\frac{1}{2}=-\frac{1}{2}$$

Subtract \frac{1}{2} from both sides of the equation.

$$b=0 b=-1$$

Solve for y (complex solution)

$$y\in \mathrm{C}$$

$$b=-1\text{ or }b=0$$

Solve for y

$$y\in \mathrm{R}$$

$$b=-1\text{ or }b=0$$