Finite Math Examples

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

Step 1

Set $- 2 {(x - 1)}^{2} + 4$ equal to $0$ .

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

Step 2

Solve for $x$ .

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Step 2.1

Simplify $- 2 {(x - 1)}^{2} + 4$ .

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

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

$- 2 ((x - 1) (x - 1)) + 4 = 0$

Step 2.1.1.2

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

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Step 2.1.1.2.1

Apply the distributive property.

$- 2 (x (x - 1) - 1 (x - 1)) + 4 = 0$

Step 2.1.1.2.2

Apply the distributive property.

$- 2 (x \cdot x + x \cdot - 1 - 1 (x - 1)) + 4 = 0$

Step 2.1.1.2.3

Apply the distributive property.

$- 2 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) + 4 = 0$

Step 2.1.1.3

Simplify and combine like terms.

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Step 2.1.1.3.1

Simplify each term.

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Step 2.1.1.3.1.1

Multiply $x$ by $x$ .

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

Step 2.1.1.3.1.2

Move $- 1$ to the left of $x$ .

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

Step 2.1.1.3.1.3

Rewrite $- 1 x$ as $- x$ .

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

Step 2.1.1.3.1.4

Rewrite $- 1 x$ as $- x$ .

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

Step 2.1.1.3.1.5

Multiply $- 1$ by $- 1$ .

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

Step 2.1.1.3.2

Subtract $x$ from $- x$ .

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

Step 2.1.1.4

Apply the distributive property.

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

Step 2.1.1.5

Simplify.

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Step 2.1.1.5.1

Multiply $- 2$ by $- 2$ .

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

Step 2.1.1.5.2

Multiply $- 2$ by $1$ .

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

Step 2.1.2

Add $- 2$ and $4$ .

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

Step 2.2

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

$x = 1 - \sqrt{2}, 1 + \sqrt{2}$

Step 3

The result can be shown in multiple forms.

Exact Form:

$x = 1 - \sqrt{2}, 1 + \sqrt{2}$

Decimal Form:

$x = - 0.41421356 \dots, 2.41421356 \dots$

Step 4