Algebra Examples

y = x^{2} - 4 x + 3

$y = x^{2} - 4 x + 3$

Step 1

Rewrite the equation in vertex form.

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Step 1.1

Complete the square for

x^{2} - 4 x + 3

$x^{2} - 4 x + 3$ .

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Step 1.1.1

Use the form

a x^{2} + b x + c

$a x^{2} + b x + c$ , to find the values of

a

$a$ ,

b

$b$ , and

c

$c$ .

a = 1

$a = 1$

b = - 4

$b = - 4$

c = 3

$c = 3$

Step 1.1.2

Consider the vertex form of a parabola.

a {(x + d)}^{2} + e

$a {(x + d)}^{2} + e$

Step 1.1.3

Find the value of

d

$d$ using the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

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Step 1.1.3.1

Substitute the values of

a

$a$ and

b

$b$ into the formula

d = \frac{b}{2 a}

$d = \frac{b}{2 a}$ .

d = \frac{- 4}{2 \cdot 1}

$d = \frac{- 4}{2 \cdot 1}$

Step 1.1.3.2

Cancel the common factor of

- 4

$- 4$ and

2

$2$ .

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Step 1.1.3.2.1

Factor

2

$2$ out of

- 4

$- 4$ .

d = \frac{2 \cdot - 2}{2 \cdot 1}

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 1.1.3.2.2

Cancel the common factors.

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Step 1.1.3.2.2.1

Factor

2

$2$ out of

2 \cdot 1

$2 \cdot 1$ .

d = \frac{2 \cdot - 2}{2 (1)}

$d = \frac{2 \cdot - 2}{2 (1)}$

Step 1.1.3.2.2.2

Cancel the common factor.

d = \frac{2 \cdot - 2}{2 \cdot 1}

$d = \frac{2 \cdot - 2}{2 \cdot 1}$

Step 1.1.3.2.2.3

Rewrite the expression.

d = \frac{- 2}{1}

$d = \frac{- 2}{1}$

Step 1.1.3.2.2.4

Divide

- 2

$- 2$ by

1

$1$ .

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

d = - 2

$d = - 2$

Step 1.1.4

Find the value of

e

$e$ using the formula

e = c - \frac{b^{2}}{4 a}

$e = c - \frac{b^{2}}{4 a}$ .

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Step 1.1.4.1

Substitute the values of

c

$c$ ,

b

$b$ and

a

$a$ into the formula

e = c - \frac{b^{2}}{4 a}

$e = c - \frac{b^{2}}{4 a}$ .

e = 3 - \frac{{(- 4)}^{2}}{4 \cdot 1}

$e = 3 - \frac{{(- 4)}^{2}}{4 \cdot 1}$

Step 1.1.4.2

Simplify the right side.

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Step 1.1.4.2.1

Simplify each term.

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Step 1.1.4.2.1.1

Cancel the common factor of

{(- 4)}^{2}

${(- 4)}^{2}$ and

4

$4$ .

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Step 1.1.4.2.1.1.1

Rewrite

- 4

$- 4$ as

- 1 (4)

$- 1 (4)$ .

e = 3 - \frac{{(- 1 (4))}^{2}}{4 \cdot 1}

$e = 3 - \frac{{(- 1 (4))}^{2}}{4 \cdot 1}$

Step 1.1.4.2.1.1.2

Apply the product rule to

- 1 (4)

$- 1 (4)$ .

e = 3 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 1}

$e = 3 - \frac{{(- 1)}^{2} \cdot 4^{2}}{4 \cdot 1}$

Step 1.1.4.2.1.1.3

Raise

- 1

$- 1$ to the power of

2

$2$ .

e = 3 - \frac{1 \cdot 4^{2}}{4 \cdot 1}

$e = 3 - \frac{1 \cdot 4^{2}}{4 \cdot 1}$

Step 1.1.4.2.1.1.4

Multiply

4^{2}

$4^{2}$ by

1

$1$ .

e = 3 - \frac{4^{2}}{4 \cdot 1}

$e = 3 - \frac{4^{2}}{4 \cdot 1}$

Step 1.1.4.2.1.1.5

Factor

4

$4$ out of

4^{2}

$4^{2}$ .

e = 3 - \frac{4 \cdot 4}{4 \cdot 1}

$e = 3 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.1.4.2.1.1.6

Cancel the common factors.

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Step 1.1.4.2.1.1.6.1

Factor

4

$4$ out of

4 \cdot 1

$4 \cdot 1$ .

e = 3 - \frac{4 \cdot 4}{4 (1)}

$e = 3 - \frac{4 \cdot 4}{4 (1)}$

Step 1.1.4.2.1.1.6.2

Cancel the common factor.

e = 3 - \frac{4 \cdot 4}{4 \cdot 1}

$e = 3 - \frac{4 \cdot 4}{4 \cdot 1}$

Step 1.1.4.2.1.1.6.3

Rewrite the expression.

e = 3 - \frac{4}{1}

$e = 3 - \frac{4}{1}$

Step 1.1.4.2.1.1.6.4

Divide

4

$4$ by

1

$1$ .

e = 3 - 1 \cdot 4

$e = 3 - 1 \cdot 4$

e = 3 - 1 \cdot 4

$e = 3 - 1 \cdot 4$

e = 3 - 1 \cdot 4

$e = 3 - 1 \cdot 4$

Step 1.1.4.2.1.2

Multiply

- 1

$- 1$ by

4

$4$ .

e = 3 - 4

$e = 3 - 4$

e = 3 - 4

$e = 3 - 4$

Step 1.1.4.2.2

Subtract

4

$4$ from

3

$3$ .

e = - 1

$e = - 1$

e = - 1

$e = - 1$

e = - 1

$e = - 1$

Step 1.1.5

Substitute the values of

a

$a$ ,

d

$d$ , and

e

$e$ into the vertex form

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

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

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

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

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

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

Step 1.2

Set

y

$y$ equal to the new right side.

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

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

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

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

Step 2

Use the vertex form,

y = a {(x - h)}^{2} + k

$y = a {(x - h)}^{2} + k$ , to determine the values of

a

$a$ ,

h

$h$ , and

k

$k$ .

a = 1

$a = 1$

h = 2

$h = 2$

k = - 1

$k = - 1$

Step 3

Since the value of

a

$a$ is positive, the parabola opens up.

Opens Up

Step 4

Find the vertex

(h, k)

$(h, k)$ .

(2, - 1)

$(2, - 1)$

Step 5

Find

p

$p$ , the distance from the vertex to the focus.

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Step 5.1

Find the distance from the vertex to a focus of the parabola by using the following formula.

\frac{1}{4 a}

$\frac{1}{4 a}$

Step 5.2

Substitute the value of

a

$a$ into the formula.

\frac{1}{4 \cdot 1}

$\frac{1}{4 \cdot 1}$

Step 5.3

Cancel the common factor of

1

$1$ .

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Step 5.3.1

Cancel the common factor.

\frac{1}{4 \cdot 1}

$\frac{1}{4 \cdot 1}$

Step 5.3.2

Rewrite the expression.

\frac{1}{4}

$\frac{1}{4}$

\frac{1}{4}

$\frac{1}{4}$

\frac{1}{4}

$\frac{1}{4}$

Step 6

Find the focus.

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Step 6.1

The focus of a parabola can be found by adding

p

$p$ to the y-coordinate

k

$k$ if the parabola opens up or down.

(h, k + p)

$(h, k + p)$

Step 6.2

Substitute the known values of

h

$h$ ,

p

$p$ , and

k

$k$ into the formula and simplify.

(2, - \frac{3}{4})

$(2, - \frac{3}{4})$

(2, - \frac{3}{4})

$(2, - \frac{3}{4})$

Step 7

Find the axis of symmetry by finding the line that passes through the vertex and the focus.

x = 2

$x = 2$

Step 8