Rewrite the equation in vertex form.

Complete the square for <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>14</mn><mi>x</mi><mo>+</mo><mn>48</mn></mstyle></math> .

Use the form <math><mstyle displaystyle="true"><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi><mi>x</mi><mo>+</mo><mi>c</mi></mstyle></math> , to find the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> .

Consider the vertex form of a parabola.

Substitute the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> into the formula <math><mstyle displaystyle="true"><mi>d</mi><mo>=</mo><mfrac><mrow><mi>b</mi></mrow><mrow><mn>2</mn><mi>a</mi></mrow></mfrac></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>14</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>14</mn></mstyle></math> .

Cancel the common factors.

Factor <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> out of <math><mstyle displaystyle="true"><mn>2</mn><mo>⋅</mo><mn>1</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Divide <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Find the value of <math><mstyle displaystyle="true"><mi>e</mi></mstyle></math> using the formula <math><mstyle displaystyle="true"><mi>e</mi><mo>=</mo><mi>c</mi><mo>-</mo><mfrac><mrow><msup><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>4</mn><mi>a</mi></mrow></mfrac></mstyle></math> .

Simplify each term.

Raise <math><mstyle displaystyle="true"><mn>14</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Divide <math><mstyle displaystyle="true"><mn>196</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>49</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>49</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>48</mn></mstyle></math> .

Substitute the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>d</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>e</mi></mstyle></math> into the vertex form <math><mstyle displaystyle="true"><mi>a</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mi>d</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>e</mi></mstyle></math> .

Set <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> equal to the new right side.

Use the vertex form, <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mi>a</mi><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mi>h</mi><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>k</mi></mstyle></math> , to determine the values of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> .

Since the value of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> is positive, the parabola opens up.

Opens Up

Find the vertex <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mstyle></math> .

Find <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> , the distance from the vertex to the focus.

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

Substitute the value of <math><mstyle displaystyle="true"><mi>a</mi></mstyle></math> into the formula.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Cancel the common factor.

Rewrite the expression.

Find the focus.

The focus of a parabola can be found by adding <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> to the y-coordinate <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> if the parabola opens up or down.

Substitute the known values of <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> , <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> , and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> into the formula and simplify.

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

Find the directrix.

The directrix of a parabola is the horizontal line found by subtracting <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> from the y-coordinate <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> of the vertex if the parabola opens up or down.

Substitute the known values of <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> into the formula and simplify.

Use the properties of the parabola to analyze and graph the parabola.

Direction: Opens Up

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>7</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>7</mn><mo>,</mo><mo>-</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>7</mn></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>

Direction: Opens Up

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>7</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>7</mn><mo>,</mo><mo>-</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>7</mn></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>14</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>8</mn></mstyle></math> .

Simplify by adding and subtracting.

Subtract <math><mstyle displaystyle="true"><mn>112</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>64</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>48</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>48</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>8</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>9</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>9</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>14</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>9</mn></mstyle></math> .

Simplify by adding and subtracting.

Subtract <math><mstyle displaystyle="true"><mn>126</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>81</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>45</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>48</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>9</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>6</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>6</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>14</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>6</mn></mstyle></math> .

Simplify by adding and subtracting.

Subtract <math><mstyle displaystyle="true"><mn>84</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>36</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>48</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>48</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>6</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn>5</mn></mstyle></math> in the expression.

Simplify the result.

Simplify each term.

Raise <math><mstyle displaystyle="true"><mo>-</mo><mn>5</mn></mstyle></math> to the power of <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mn>14</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>5</mn></mstyle></math> .

Simplify by adding and subtracting.

Subtract <math><mstyle displaystyle="true"><mn>70</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>25</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>45</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>48</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

The <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> value at <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>5</mn></mstyle></math> is <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Graph the parabola using its properties and the selected points.

Graph the parabola using its properties and the selected points.

Direction: Opens Up

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>7</mn><mo>,</mo><mo>-</mo><mn>1</mn><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn>7</mn><mo>,</mo><mo>-</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mn>7</mn></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>5</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math>

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Name | one billion six hundred two million four hundred twenty-five thousand sixty-one |
---|

- 1602425061 has 8 divisors, whose sum is
**2159540960** - The reverse of 1602425061 is
**1605242061** - Previous prime number is
**3**

- Is Prime? no
- Number parity odd
- Number length 10
- Sum of Digits 27
- Digital Root 9

Name | one billion four hundred ten million six hundred forty-seven thousand eight hundred sixty-eight |
---|

- 1410647868 has 64 divisors, whose sum is
**3705730560** - The reverse of 1410647868 is
**8687460141** - Previous prime number is
**7**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 45
- Digital Root 9

Name | one billion four hundred ninety-five million eight hundred forty-five thousand three hundred forty |
---|

- 1495845340 has 32 divisors, whose sum is
**4405945104** - The reverse of 1495845340 is
**0435485941** - Previous prime number is
**5**

- Is Prime? no
- Number parity even
- Number length 10
- Sum of Digits 43
- Digital Root 7