Calculus Examples

$7 x {(2 x - 1)}^{2} = 1$

Step 1

Simplify $7 x {(2 x - 1)}^{2}$ .

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Step 1.1

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

$7 x ((2 x - 1) (2 x - 1)) = 1$

Step 1.2

Expand $(2 x - 1) (2 x - 1)$ using the FOIL Method.

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Step 1.2.1

Apply the distributive property.

$7 x (2 x (2 x - 1) - 1 (2 x - 1)) = 1$

Step 1.2.2

Apply the distributive property.

$7 x (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x - 1)) = 1$

Step 1.2.3

Apply the distributive property.

$7 x (2 x (2 x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) = 1$

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.

$7 x (2 \cdot 2 x \cdot x + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) = 1$

Step 1.3.1.2

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

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Step 1.3.1.2.1

Move $x$ .

$7 x (2 \cdot 2 (x \cdot x) + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) = 1$

Step 1.3.1.2.2

Multiply $x$ by $x$ .

$7 x (2 \cdot 2 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) = 1$

Step 1.3.1.3

Multiply $2$ by $2$ .

$7 x (4 x^{2} + 2 x \cdot - 1 - 1 (2 x) - 1 \cdot - 1) = 1$

Step 1.3.1.4

Multiply $- 1$ by $2$ .

$7 x (4 x^{2} - 2 x - 1 (2 x) - 1 \cdot - 1) = 1$

Step 1.3.1.5

Multiply $2$ by $- 1$ .

$7 x (4 x^{2} - 2 x - 2 x - 1 \cdot - 1) = 1$

Step 1.3.1.6

Multiply $- 1$ by $- 1$ .

$7 x (4 x^{2} - 2 x - 2 x + 1) = 1$

Step 1.3.2

Subtract $2 x$ from $- 2 x$ .

$7 x (4 x^{2} - 4 x + 1) = 1$

Step 1.4

Apply the distributive property.

$7 x (4 x^{2}) + 7 x (- 4 x) + 7 x \cdot 1 = 1$

Step 1.5

Simplify.

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Step 1.5.1

Rewrite using the commutative property of multiplication.

$7 \cdot 4 x \cdot x^{2} + 7 x (- 4 x) + 7 x \cdot 1 = 1$

Step 1.5.2

Rewrite using the commutative property of multiplication.

$7 \cdot 4 x \cdot x^{2} + 7 \cdot - 4 x \cdot x + 7 x \cdot 1 = 1$

Step 1.5.3

Multiply $7$ by $1$ .

$7 \cdot 4 x \cdot x^{2} + 7 \cdot - 4 x \cdot x + 7 x = 1$

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 $x^{2}$ .

$7 \cdot 4 (x^{2} x) + 7 \cdot - 4 x \cdot x + 7 x = 1$

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 $1$ .

$7 \cdot 4 (x^{2} x^{1}) + 7 \cdot - 4 x \cdot x + 7 x = 1$

Step 1.6.1.2.2

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

$7 \cdot 4 x^{2 + 1} + 7 \cdot - 4 x \cdot x + 7 x = 1$

Step 1.6.1.3

Add $2$ and $1$ .

$7 \cdot 4 x^{3} + 7 \cdot - 4 x \cdot x + 7 x = 1$

Step 1.6.2

Multiply $7$ by $4$ .

$28 x^{3} + 7 \cdot - 4 x \cdot x + 7 x = 1$

Step 1.6.3

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

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Step 1.6.3.1

Move $x$ .

$28 x^{3} + 7 \cdot - 4 (x \cdot x) + 7 x = 1$

Step 1.6.3.2

Multiply $x$ by $x$ .

$28 x^{3} + 7 \cdot - 4 x^{2} + 7 x = 1$

Step 1.6.4

Multiply $7$ by $- 4$ .

$28 x^{3} - 28 x^{2} + 7 x = 1$

Step 2

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

$x \approx 0.72235444$

Step 3