Calculus Examples

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

Step 1

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

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Step 1.1

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

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

Step 1.2

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

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Step 1.2.1

Apply the distributive property.

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

Step 1.2.2

Apply the distributive property.

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

Step 1.2.3

Apply the distributive property.

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

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

Multiply $x$ by $x$ .

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

Step 1.3.1.2

Move $- 2$ to the left of $x$ .

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

Step 1.3.1.3

Multiply $- 2$ by $- 2$ .

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

Step 1.3.2

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

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

Step 1.4

Apply the distributive property.

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

Step 1.5

Simplify.

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Step 1.5.1

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

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Step 1.5.1.1

Move $x^{2}$ .

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

Step 1.5.1.2

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

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Step 1.5.1.2.1

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

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

Step 1.5.1.2.2

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

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

Step 1.5.1.3

Add $2$ and $1$ .

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

Step 1.5.2

Rewrite using the commutative property of multiplication.

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

Step 1.5.3

Multiply $4$ by $7$ .

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

Step 1.6

Simplify each term.

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Step 1.6.1

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

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Step 1.6.1.1

Move $x$ .

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

Step 1.6.1.2

Multiply $x$ by $x$ .

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

Step 1.6.2

Multiply $7$ by $- 4$ .

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

Step 2

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

$x \approx 0.03707618, 1.71105223, 2.25187158$

Step 3