Calculus Examples

$x {(5 x - 1)}^{2} = 1$

Step 1

Simplify $x {(5 x - 1)}^{2}$ .

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Step 1.1

Rewrite ${(5 x - 1)}^{2}$ as $(5 x - 1) (5 x - 1)$ .

$x ((5 x - 1) (5 x - 1)) = 1$

Step 1.2

Expand $(5 x - 1) (5 x - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$x (5 x (5 x - 1) - 1 (5 x - 1)) = 1$

Step 1.2.2

Apply the distributive property.

$x (5 x (5 x) + 5 x \cdot - 1 - 1 (5 x - 1)) = 1$

Step 1.2.3

Apply the distributive property.

$x (5 x (5 x) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) = 1$

Step 1.3

Simplify and combine like terms.

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Step 1.3.1

Simplify each term.

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Step 1.3.1.1

Rewrite using the commutative property of multiplication.

$x (5 \cdot 5 x \cdot x + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) = 1$

Step 1.3.1.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.3.1.2.1

Move $x$ .

$x (5 \cdot 5 (x \cdot x) + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) = 1$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

$x (5 \cdot 5 x^{2} + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) = 1$

Step 1.3.1.3

Multiply $5$ by $5$ .

$x (25 x^{2} + 5 x \cdot - 1 - 1 (5 x) - 1 \cdot - 1) = 1$

Step 1.3.1.4

Multiply $- 1$ by $5$ .

$x (25 x^{2} - 5 x - 1 (5 x) - 1 \cdot - 1) = 1$

Step 1.3.1.5

Multiply $5$ by $- 1$ .

$x (25 x^{2} - 5 x - 5 x - 1 \cdot - 1) = 1$

Step 1.3.1.6

Multiply $- 1$ by $- 1$ .

$x (25 x^{2} - 5 x - 5 x + 1) = 1$

Step 1.3.2

Subtract $5 x$ from $- 5 x$ .

$x (25 x^{2} - 10 x + 1) = 1$

Step 1.4

Apply the distributive property.

$x (25 x^{2}) + x (- 10 x) + x \cdot 1 = 1$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite using the commutative property of multiplication.

$25 x \cdot x^{2} + x (- 10 x) + x \cdot 1 = 1$

Step 1.5.2

Rewrite using the commutative property of multiplication.

$25 x \cdot x^{2} - 10 x \cdot x + x \cdot 1 = 1$

Step 1.5.3

Multiply $x$ by $1$ .

$25 x \cdot x^{2} - 10 x \cdot x + x = 1$

Step 1.6

Simplify each term.

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Step 1.6.1

Multiply $x$ by $x^{2}$ by adding the exponents.

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Step 1.6.1.1

Move $x^{2}$ .

$25 (x^{2} x) - 10 x \cdot x + x = 1$

Step 1.6.1.2

Multiply $x^{2}$ by $x$ .

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Step 1.6.1.2.1

Raise $x$ to the power of $1$ .

$25 (x^{2} x^{1}) - 10 x \cdot x + x = 1$

Step 1.6.1.2.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$25 x^{2 + 1} - 10 x \cdot x + x = 1$

Step 1.6.1.3

Add $2$ and $1$ .

$25 x^{3} - 10 x \cdot x + x = 1$

Step 1.6.2

Multiply $x$ by $x$ by adding the exponents.

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Step 1.6.2.1

Move $x$ .

$25 x^{3} - 10 (x \cdot x) + x = 1$

Step 1.6.2.2

Multiply $x$ by $x$ .

$25 x^{3} - 10 x^{2} + x = 1$

Step 2

Graph each side of the equation. The solution is the x-value of the point of intersection.

$x \approx 0.48668553$

Step 3