Algebra Examples

$3 x^{2} + 4 x = y$ , $[0, 100]$

Step 1

Rewrite the equation as $y = 3 x^{2} + 4 x$ .

$y = 3 x^{2} + 4 x$

Step 2

The Intermediate Value Theorem states that, if $f$ is a real-valued continuous function on the interval $[a, b]$ , and $u$ is a number between $f (a)$ and $f (b)$ , then there is a $c$ contained in the interval $[a, b]$ such that $f (c) = u$ .

$u = f (c) = 0$

Step 3

The domain of the expression is all real numbers except where the expression is undefined. In this case, there is no real number that makes the expression undefined.

Interval Notation:

$(- \infty, \infty)$

Set-Builder Notation:

${x | x \in ℝ}$

Step 4

Calculate $f (a) = f (0) = 3 {(0)}^{2} + 4 (0)$ .

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Step 4.1

Simplify each term.

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Step 4.1.1

Raising $0$ to any positive power yields $0$ .

$f (0) = 3 \cdot 0 + 4 (0)$

Step 4.1.2

Multiply $3$ by $0$ .

$f (0) = 0 + 4 (0)$

Step 4.1.3

Multiply $4$ by $0$ .

$f (0) = 0 + 0$

Step 4.2

Add $0$ and $0$ .

$f (0) = 0$

Step 5

Calculate $f (b) = f (100) = 3 {(100)}^{2} + 4 (100)$ .

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Step 5.1

Simplify each term.

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Step 5.1.1

Raise $100$ to the power of $2$ .

$f (100) = 3 \cdot 10000 + 4 (100)$

Step 5.1.2

Multiply $3$ by $10000$ .

$f (100) = 30000 + 4 (100)$

Step 5.1.3

Multiply $4$ by $100$ .

$f (100) = 30000 + 400$

Step 5.2

Add $30000$ and $400$ .

$f (100) = 30400$

Step 6

Since $0$ is on the interval $[0, 30400]$ , solve the equation for $x$ at the root by setting $y$ to $0$ in $y = 3 x^{2} + 4 x$ .

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Step 6.1

Rewrite the equation as $3 x^{2} + 4 x = 0$ .

$3 x^{2} + 4 x = 0$

Step 6.2

Factor $x$ out of $3 x^{2} + 4 x$ .

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Step 6.2.1

Factor $x$ out of $3 x^{2}$ .

$x (3 x) + 4 x = 0$

Step 6.2.2

Factor $x$ out of $4 x$ .

$x (3 x) + x \cdot 4 = 0$

Step 6.2.3

Factor $x$ out of $x (3 x) + x \cdot 4$ .

$x (3 x + 4) = 0$

Step 6.3

If any individual factor on the left side of the equation is equal to $0$ , the entire expression will be equal to $0$ .

$x = 0$

$3 x + 4 = 0$

Step 6.4

Set $x$ equal to $0$ .

$x = 0$

Step 6.5

Set $3 x + 4$ equal to $0$ and solve for $x$ .

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Step 6.5.1

Set $3 x + 4$ equal to $0$ .

$3 x + 4 = 0$

Step 6.5.2

Solve $3 x + 4 = 0$ for $x$ .

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Step 6.5.2.1

Subtract $4$ from both sides of the equation.

$3 x = - 4$

Step 6.5.2.2

Divide each term in $3 x = - 4$ by $3$ and simplify.

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Step 6.5.2.2.1

Divide each term in $3 x = - 4$ by $3$ .

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 6.5.2.2.2

Simplify the left side.

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Step 6.5.2.2.2.1

Cancel the common factor of $3$ .

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Step 6.5.2.2.2.1.1

Cancel the common factor.

$\frac{3 x}{3} = \frac{- 4}{3}$

Step 6.5.2.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{- 4}{3}$

Step 6.5.2.2.3

Simplify the right side.

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Step 6.5.2.2.3.1

Move the negative in front of the fraction.

$x = - \frac{4}{3}$

Step 6.6

The final solution is all the values that make $x (3 x + 4) = 0$ true.

$x = 0, - \frac{4}{3}$

Step 7

The Intermediate Value Theorem states that there is a root $f (c) = 0$ on the interval $[0, 30400]$ because $f$ is a continuous function on $[0, 100]$ .

The roots on the interval $[0, 100]$ are located at $x = 0$ .

Step 8