Calculus Examples

$f (x) = 4 x^{2} + 2$ , $f (x) = 4 x + 1$

Step 1

Substitute $4 x + 1$ for $f (x)$ .

$4 x + 1 = 4 x^{2} + 2$

Step 2

Solve for $x$ .

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Step 2.1

Subtract $4 x^{2}$ from both sides of the equation.

$4 x + 1 - 4 x^{2} = 2$

Step 2.2

Subtract $2$ from both sides of the equation.

$4 x + 1 - 4 x^{2} - 2 = 0$

Step 2.3

Subtract $2$ from $1$ .

$4 x - 4 x^{2} - 1 = 0$

Step 2.4

Factor the left side of the equation.

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Step 2.4.1

Factor $- 1$ out of $4 x - 4 x^{2} - 1$ .

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Step 2.4.1.1

Reorder $4 x$ and $- 4 x^{2}$ .

$- 4 x^{2} + 4 x - 1 = 0$

Step 2.4.1.2

Factor $- 1$ out of $- 4 x^{2}$ .

$- (4 x^{2}) + 4 x - 1 = 0$

Step 2.4.1.3

Factor $- 1$ out of $4 x$ .

$- (4 x^{2}) - (- 4 x) - 1 = 0$

Step 2.4.1.4

Rewrite $- 1$ as $- 1 (1)$ .

$- (4 x^{2}) - (- 4 x) - 1 \cdot 1 = 0$

Step 2.4.1.5

Factor $- 1$ out of $- (4 x^{2}) - (- 4 x)$ .

$- (4 x^{2} - 4 x) - 1 \cdot 1 = 0$

Step 2.4.1.6

Factor $- 1$ out of $- (4 x^{2} - 4 x) - 1 (1)$ .

$- (4 x^{2} - 4 x + 1) = 0$

Step 2.4.2

Factor using the perfect square rule.

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Step 2.4.2.1

Rewrite $4 x^{2}$ as ${(2 x)}^{2}$ .

$- ({(2 x)}^{2} - 4 x + 1) = 0$

Step 2.4.2.2

Rewrite $1$ as $1^{2}$ .

$- ({(2 x)}^{2} - 4 x + 1^{2}) = 0$

Step 2.4.2.3

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$4 x = 2 \cdot (2 x) \cdot 1$

Step 2.4.2.4

Rewrite the polynomial.

$- ({(2 x)}^{2} - 2 \cdot (2 x) \cdot 1 + 1^{2}) = 0$

Step 2.4.2.5

Factor using the perfect square trinomial rule $a^{2} - 2 a b + b^{2} = {(a - b)}^{2}$ , where $a = 2 x$ and $b = 1$ .

$- {(2 x - 1)}^{2} = 0$

Step 2.5

Divide each term in $- {(2 x - 1)}^{2} = 0$ by $- 1$ and simplify.

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Step 2.5.1

Divide each term in $- {(2 x - 1)}^{2} = 0$ by $- 1$ .

$\frac{- {(2 x - 1)}^{2}}{- 1} = \frac{0}{- 1}$

Step 2.5.2

Simplify the left side.

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Step 2.5.2.1

Dividing two negative values results in a positive value.

$\frac{{(2 x - 1)}^{2}}{1} = \frac{0}{- 1}$

Step 2.5.2.2

Divide ${(2 x - 1)}^{2}$ by $1$ .

${(2 x - 1)}^{2} = \frac{0}{- 1}$

Step 2.5.3

Simplify the right side.

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Step 2.5.3.1

Divide $0$ by $- 1$ .

${(2 x - 1)}^{2} = 0$

Step 2.6

Set the $2 x - 1$ equal to $0$ .

$2 x - 1 = 0$

Step 2.7

Solve for $x$ .

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Step 2.7.1

Add $1$ to both sides of the equation.

$2 x = 1$

Step 2.7.2

Divide each term in $2 x = 1$ by $2$ and simplify.

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Step 2.7.2.1

Divide each term in $2 x = 1$ by $2$ .

$\frac{2 x}{2} = \frac{1}{2}$

Step 2.7.2.2

Simplify the left side.

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Step 2.7.2.2.1

Cancel the common factor of $2$ .

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Step 2.7.2.2.1.1

Cancel the common factor.

$\frac{2 x}{2} = \frac{1}{2}$

Step 2.7.2.2.1.2

Divide $x$ by $1$ .

$x = \frac{1}{2}$

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