Algebra Examples

y = 4 {(x - 1)}^{2} - 4

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

Step 1

Set

4 {(x - 1)}^{2} - 4

$4 {(x - 1)}^{2} - 4$ equal to

0

$0$ .

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

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

Step 2

Solve for

x

$x$ .

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Step 2.1

Simplify

4 {(x - 1)}^{2} - 4

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

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Step 2.1.1

Simplify each term.

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Step 2.1.1.1

Rewrite

{(x - 1)}^{2}

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

(x - 1) (x - 1)

$(x - 1) (x - 1)$ .

4 ((x - 1) (x - 1)) - 4 = 0

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

Step 2.1.1.2

Expand

(x - 1) (x - 1)

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

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Step 2.1.1.2.1

Apply the distributive property.

4 (x (x - 1) - 1 (x - 1)) - 4 = 0

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

Step 2.1.1.2.2

Apply the distributive property.

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

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

Step 2.1.1.2.3

Apply the distributive property.

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

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

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

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

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

$x$ by

x

$x$ .

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

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

Step 2.1.1.3.1.2

Move

- 1

$- 1$ to the left of

x

$x$ .

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

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

Step 2.1.1.3.1.3

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

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

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

Step 2.1.1.3.1.4

Rewrite

- 1 x

$- 1 x$ as

- x

$- x$ .

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

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

Step 2.1.1.3.1.5

Multiply

- 1

$- 1$ by

- 1

$- 1$ .

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

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

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

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

Step 2.1.1.3.2

Subtract

x

$x$ from

- x

$- x$ .

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

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

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

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

Step 2.1.1.4

Apply the distributive property.

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

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

Step 2.1.1.5

Simplify.

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Step 2.1.1.5.1

Multiply

- 2

$- 2$ by

4

$4$ .

4 x^{2} - 8 x + 4 \cdot 1 - 4 = 0

$4 x^{2} - 8 x + 4 \cdot 1 - 4 = 0$

Step 2.1.1.5.2

Multiply

4

$4$ by

1

$1$ .

4 x^{2} - 8 x + 4 - 4 = 0

$4 x^{2} - 8 x + 4 - 4 = 0$

4 x^{2} - 8 x + 4 - 4 = 0

$4 x^{2} - 8 x + 4 - 4 = 0$

4 x^{2} - 8 x + 4 - 4 = 0

$4 x^{2} - 8 x + 4 - 4 = 0$

Step 2.1.2

Combine the opposite terms in

4 x^{2} - 8 x + 4 - 4

$4 x^{2} - 8 x + 4 - 4$ .

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Step 2.1.2.1

Subtract

4

$4$ from

4

$4$ .

4 x^{2} - 8 x + 0 = 0

$4 x^{2} - 8 x + 0 = 0$

Step 2.1.2.2

Add

4 x^{2} - 8 x

$4 x^{2} - 8 x$ and

0

$0$ .

4 x^{2} - 8 x = 0

$4 x^{2} - 8 x = 0$

4 x^{2} - 8 x = 0

$4 x^{2} - 8 x = 0$

4 x^{2} - 8 x = 0

$4 x^{2} - 8 x = 0$

Step 2.2

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

x = 0, 2

$x = 0, 2$

x = 0, 2

$x = 0, 2$

Step 3