Algebra Examples

$x = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}$

Step 1

Move all terms to the left side of the equation and simplify.

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Step 1.1

Simplify the right side.

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Step 1.1.1

Simplify $\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}$ .

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Step 1.1.1.1

Split the fraction $\frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}$ into two fractions.

$x = \frac{- b}{2 a} + \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

Step 1.1.1.2

Move the negative in front of the fraction.

$x = - \frac{b}{2 a} + \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

Step 1.2

Move all the expressions to the left side of the equation.

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Step 1.2.1

Add $\frac{b}{2 a}$ to both sides of the equation.

$x + \frac{b}{2 a} = \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

Step 1.2.2

Subtract $\frac{\sqrt{b^{2} - 4 a c}}{2 a}$ from both sides of the equation.

$x + \frac{b}{2 a} - \frac{\sqrt{b^{2} - 4 a c}}{2 a} = 0$

Step 2

Simplify $x + \frac{b}{2 a} - \frac{\sqrt{b^{2} - 4 a c}}{2 a}$ .

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Step 2.1

To write $x$ as a fraction with a common denominator, multiply by $\frac{2 a}{2 a}$ .

$x \cdot \frac{2 a}{2 a} - \frac{\sqrt{b^{2} - 4 a c}}{2 a} + \frac{b}{2 a} = 0$

Step 2.2

Simplify terms.

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Step 2.2.1

Combine $x$ and $\frac{2 a}{2 a}$ .

$\frac{x (2 a)}{2 a} - \frac{\sqrt{b^{2} - 4 a c}}{2 a} + \frac{b}{2 a} = 0$

Step 2.2.2

Combine the numerators over the common denominator.

$\frac{x (2 a) - \sqrt{b^{2} - 4 a c}}{2 a} + \frac{b}{2 a} = 0$

Step 2.3

Move $2$ to the left of $x$ .

$\frac{2 x a - \sqrt{b^{2} - 4 a c}}{2 a} + \frac{b}{2 a} = 0$

Step 3

Subtract $\frac{b}{2 a}$ from both sides of the equation.

$\frac{2 x a - \sqrt{b^{2} - 4 a c}}{2 a} = - \frac{b}{2 a}$

Step 4

Since the expression on each side of the equation has the same denominator, the numerators must be equal.

$2 x a - \sqrt{b^{2} - 4 a c} = - b$

Step 5

Add $\sqrt{b^{2} - 4 a c}$ to both sides of the equation.

$2 x a = - b + \sqrt{b^{2} - 4 a c}$

Step 6

Divide each term in $2 x a = - b + \sqrt{b^{2} - 4 a c}$ by $2 a$ and simplify.

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Step 6.1

Divide each term in $2 x a = - b + \sqrt{b^{2} - 4 a c}$ by $2 a$ .

$\frac{2 x a}{2 a} = \frac{- b}{2 a} + \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

Step 6.2

Simplify the left side.

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Step 6.2.1

Cancel the common factor of $2$ .

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Step 6.2.1.1

Cancel the common factor.

$\frac{2 x a}{2 a} = \frac{- b}{2 a} + \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

Step 6.2.1.2

Rewrite the expression.

$\frac{x a}{a} = \frac{- b}{2 a} + \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

Step 6.2.2

Cancel the common factor of $a$ .

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Step 6.2.2.1

Cancel the common factor.

$\frac{x a}{a} = \frac{- b}{2 a} + \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

Step 6.2.2.2

Divide $x$ by $1$ .

$x = \frac{- b}{2 a} + \frac{\sqrt{b^{2} - 4 a c}}{2 a}$

Step 6.3

Simplify the right side.

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Step 6.3.1

Combine the numerators over the common denominator.

$x = \frac{- b + \sqrt{b^{2} - 4 a c}}{2 a}$

Step 6.3.2

Factor $- 1$ out of $- b$ .

$x = \frac{- (b) + \sqrt{b^{2} - 4 a c}}{2 a}$

Step 6.3.3

Factor $- 1$ out of $\sqrt{b^{2} - 4 a c}$ .

$x = \frac{- (b) - 1 (- \sqrt{b^{2} - 4 a c})}{2 a}$

Step 6.3.4

Factor $- 1$ out of $- (b) - 1 (- \sqrt{b^{2} - 4 a c})$ .

$x = \frac{- (b - \sqrt{b^{2} - 4 a c})}{2 a}$

Step 6.3.5

Simplify the expression.

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Step 6.3.5.1

Rewrite $- (b - \sqrt{b^{2} - 4 a c})$ as $- 1 (b - \sqrt{b^{2} - 4 a c})$ .

$x = \frac{- 1 (b - \sqrt{b^{2} - 4 a c})}{2 a}$

Step 6.3.5.2

Move the negative in front of the fraction.

$x = - \frac{b - \sqrt{b^{2} - 4 a c}}{2 a}$