Solve alpha+beta=frac{-b+sqrt{d}}{2a}+frac{-b-sqrt{d}}{2a}

$$\alpha + \beta = \frac { - b + \sqrt { d } } { 2 a } + \frac { - b - \sqrt { d } } { 2 a }$$

Solve for a

$$\left\{\begin{matrix}a=-\frac{b}{\alpha +\beta }\text{, }&b\neq 0\text{ and }\alpha \neq -\beta \text{ and }d\geq 0\\a\neq 0\text{, }&\alpha =-\beta \text{ and }b=0\text{ and }d\geq 0\end{matrix}\right.$$

Steps for Solving Linear Equation

Variable a cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 2a.

$$2a\alpha +2a\beta =-b+\sqrt{d}-b-\sqrt{d}$$

Combine -b and -b to get 2\left(-b\right).

$$2a\alpha +2a\beta =2\left(-b\right)+\sqrt{d}-\sqrt{d}$$

Combine \sqrt{d} and -\sqrt{d} to get 0.

$$2a\alpha +2a\beta =2\left(-b\right)$$

Multiply 2 and -1 to get -2.

$$2a\alpha +2a\beta =-2b$$

Combine all terms containing a.

$$\left(2\alpha +2\beta \right)a=-2b$$

Divide both sides by 2\alpha +2\beta .

$$\frac{\left(2\alpha +2\beta \right)a}{2\alpha +2\beta }=-\frac{2b}{2\alpha +2\beta }$$

Dividing by 2\alpha +2\beta undoes the multiplication by 2\alpha +2\beta .

$$a=-\frac{2b}{2\alpha +2\beta }$$

Divide -2b by 2\alpha +2\beta .

$$a=-\frac{b}{\alpha +\beta }$$

Variable a cannot be equal to 0.

$$a=-\frac{b}{\alpha +\beta }\text{, }a\neq 0$$

Solve for b

$$b=-a\left(\alpha +\beta \right)$$

$$d\geq 0\text{ and }a\neq 0$$

Steps for Solving Linear Equation

Multiply both sides of the equation by 2a.

$$2a\alpha +2a\beta =-b+\sqrt{d}-b-\sqrt{d}$$

Combine -b and -b to get 2\left(-b\right).

$$2a\alpha +2a\beta =2\left(-b\right)+\sqrt{d}-\sqrt{d}$$

Combine \sqrt{d} and -\sqrt{d} to get 0.

$$2a\alpha +2a\beta =2\left(-b\right)$$

Swap sides so that all variable terms are on the left hand side.

$$2\left(-b\right)=2a\alpha +2a\beta$$

Multiply 2 and -1 to get -2.

$$-2b=2a\alpha +2a\beta$$

Divide both sides by -2.

$$\frac{-2b}{-2}=\frac{2a\left(\alpha +\beta \right)}{-2}$$

Dividing by -2 undoes the multiplication by -2.

$$b=\frac{2a\left(\alpha +\beta \right)}{-2}$$

Divide 2a\left(\alpha +\beta \right) by -2.

$$b=-a\left(\alpha +\beta \right)$$