Solve Using the Quadratic Formula x=(-b+- square root of b^2-4ac)/(2a)

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

Step 1

Replace all occurrences of $+$ $-$ with a single $-$ . A plus sign followed by a minus sign has the same mathematical meaning as a single minus sign because $1 \cdot - 1 = - 1$

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

Step 2

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

Simplify the right side.

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

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

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

Simplify each term.

Move the negative in front of the fraction.

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

Move the negative in front of the fraction.

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

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

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

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

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

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

Step 3

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

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

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

Simplify terms.

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

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

Combine the numerators over the common denominator.

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

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 4

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 5

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 6

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

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

Step 7

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

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

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

Simplify the left side.

Cancel the common factor of $2$ .

Cancel the common factor.

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

Rewrite the expression.

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

Cancel the common factor of $a$ .

Cancel the common factor.

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

Divide $x$ by $1$ .

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

Simplify the right side.

Combine the numerators over the common denominator.

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

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

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

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

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

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

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

Simplify the expression.

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

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

Move the negative in front of the fraction.

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

Solve Using the Quadratic Formula x=(-b+- square root of b^2-4ac)/(2a)

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