Linear Algebra Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Simplify the expression.

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Step 1.4.1

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 1.4.2

Move the negative in front of the fraction.

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

Step 2

Set the radicand in $\sqrt{b^{2} - 4 a c}$ greater than or equal to $0$ to find where the expression is defined.

$b^{2} - 4 a c \geq 0$

Step 3

Solve for $b$ .

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Step 3.1

Add $4 a c$ to both sides of the inequality.

$b^{2} \geq 4 a c$

Step 3.2

Take the specified root of both sides of the inequality to eliminate the exponent on the left side.

$\sqrt{b^{2}} \geq \sqrt{4 a c}$

Step 3.3

Simplify the equation.

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Step 3.3.1

Simplify the left side.

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Step 3.3.1.1

Pull terms out from under the radical.

$| b | \geq \sqrt{4 a c}$

Step 3.3.2

Simplify the right side.

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Step 3.3.2.1

Simplify $\sqrt{4 a c}$ .

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Step 3.3.2.1.1

Rewrite $4 a c$ as $2^{2} (a c)$ .

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Step 3.3.2.1.1.1

Rewrite $4$ as $2^{2}$ .

$| b | \geq \sqrt{2^{2} a c}$

Step 3.3.2.1.1.2

Add parentheses.

$| b | \geq \sqrt{2^{2} (a c)}$

Step 3.3.2.1.2

Pull terms out from under the radical.

$| b | \geq | 2 | \sqrt{a c}$

Step 3.3.2.1.3

The absolute value is the distance between a number and zero. The distance between $0$ and $2$ is $2$ .

$| b | \geq 2 \sqrt{a c}$

Step 3.4

Write $| b | \geq 2 \sqrt{a c}$ as a piecewise.

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Step 3.4.1

To find the interval for the first piece, find where the inside of the absolute value is non-negative.

$b \geq 0$

Step 3.4.2

In the piece where $b$ is non-negative, remove the absolute value.

$b \geq 2 \sqrt{a c}$

Step 3.4.3

Find the domain of $b \geq 2 \sqrt{a c}$ and find the intersection with $b \geq 0$ .

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Step 3.4.3.1

Find the domain of $b \geq 2 \sqrt{a c}$ .

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Step 3.4.3.1.1

Set the radicand in $\sqrt{a c}$ greater than or equal to $0$ to find where the expression is defined.

$a c \geq 0$

Step 3.4.3.1.2

Divide each term in $a c \geq 0$ by $c$ and simplify.

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Step 3.4.3.1.2.1

Divide each term in $a c \geq 0$ by $c$ .

$\frac{a c}{c} \geq \frac{0}{c}$

Step 3.4.3.1.2.2

Simplify the left side.

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Step 3.4.3.1.2.2.1

Cancel the common factor of $c$ .

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Step 3.4.3.1.2.2.1.1

Cancel the common factor.

$\frac{a c}{c} \geq \frac{0}{c}$

Step 3.4.3.1.2.2.1.2

Divide $a$ by $1$ .

$a \geq \frac{0}{c}$

Step 3.4.3.1.2.3

Simplify the right side.

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Step 3.4.3.1.2.3.1

Divide $0$ by $c$ .

$a \geq 0$

Step 3.4.3.1.3

The domain is all values of $b$ that make the expression defined.

$[0, \infty)$

Step 3.4.3.2

Find the intersection of $b \geq 0$ and $[0, \infty)$ .

$b \geq 0$

Step 3.4.4

To find the interval for the second piece, find where the inside of the absolute value is negative.

$b < 0$

Step 3.4.5

In the piece where $b$ is negative, remove the absolute value and multiply by $- 1$ .

$- b \geq 2 \sqrt{a c}$

Step 3.4.6

Find the domain of $- b \geq 2 \sqrt{a c}$ and find the intersection with $b < 0$ .

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Step 3.4.6.1

Find the domain of $- b \geq 2 \sqrt{a c}$ .

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Step 3.4.6.1.1

Set the radicand in $\sqrt{a c}$ greater than or equal to $0$ to find where the expression is defined.

$a c \geq 0$

Step 3.4.6.1.2

Divide each term in $a c \geq 0$ by $c$ and simplify.

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Step 3.4.6.1.2.1

Divide each term in $a c \geq 0$ by $c$ .

$\frac{a c}{c} \geq \frac{0}{c}$

Step 3.4.6.1.2.2

Simplify the left side.

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Step 3.4.6.1.2.2.1

Cancel the common factor of $c$ .

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Step 3.4.6.1.2.2.1.1

Cancel the common factor.

$\frac{a c}{c} \geq \frac{0}{c}$

Step 3.4.6.1.2.2.1.2

Divide $a$ by $1$ .

$a \geq \frac{0}{c}$

Step 3.4.6.1.2.3

Simplify the right side.

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Step 3.4.6.1.2.3.1

Divide $0$ by $c$ .

$a \geq 0$

Step 3.4.6.1.3

The domain is all values of $b$ that make the expression defined.

$[0, \infty)$

Step 3.4.6.2

Find the intersection of $b < 0$ and $[0, \infty)$ .

$No solution$

Step 3.4.7

Write as a piecewise.

${\begin{cases} b \geq 2 \sqrt{a c} & b \geq 0 \end{cases}$

Step 3.5

Find the intersection of $b \geq 2 \sqrt{a c}$ and $b \geq 0$ .

$b \geq 2 \sqrt{a c}$ and $b \geq 0$

Step 3.6

Find the union of the solutions.

$b \geq N o (Max i m u m)$

Step 4

Set the denominator in $\frac{b - \sqrt{b^{2} - 4 a c}}{2 a}$ equal to $0$ to find where the expression is undefined.

$2 a = 0$

Step 5

Divide each term in $2 a = 0$ by $2$ and simplify.

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Step 5.1

Divide each term in $2 a = 0$ by $2$ .

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

Step 5.2

Simplify the left side.

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Step 5.2.1

Cancel the common factor of $2$ .

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Step 5.2.1.1

Cancel the common factor.

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

Step 5.2.1.2

Divide $a$ by $1$ .

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

Step 5.3

Simplify the right side.

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Step 5.3.1

Divide $0$ by $2$ .

$a = 0$

Step 6

The domain is all values of $b$ that make the expression defined.

Interval Notation:

$(- \infty, 0) \cup (0, \infty)$

Set-Builder Notation:

${b | b \neq 0}$

Step 7