Graph b^2-4ac<0

$b^{2} - 4 a c < 0$

Step 1

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

$b^{2} < 4 a c$

Step 2

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

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

Step 3

Simplify the equation.

Simplify the left side.

Pull terms out from under the radical.

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

Simplify the right side.

Simplify $\sqrt{4 a c}$ .

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

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

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

Add parentheses.

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

Pull terms out from under the radical.

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

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

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

Step 4

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

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

$b \geq 0$

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

$b < 2 \sqrt{a c}$

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

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

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

$a c \geq 0$

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

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

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

Simplify the left side.

Cancel the common factor of $c$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

Divide $0$ by $c$ .

$a \geq 0$

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

$[0, \infty)$

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

$b \geq 0$

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

$b < 0$

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

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

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

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

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

$a c \geq 0$

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

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

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

Simplify the left side.

Cancel the common factor of $c$ .

Cancel the common factor.

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

Divide $a$ by $1$ .

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

Simplify the right side.

Divide $0$ by $c$ .

$a \geq 0$

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

$[0, \infty)$

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

$No solution$

Write as a piecewise.

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

Step 5

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

$0 \leq b < 2 \sqrt{a c}$

Step 6

Find the union of the solutions.

$0 \leq b < 2 \sqrt{a c}$

Step 7

Graph b^2-4ac<0

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