Rewrite the equation as <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> from both sides of the equation.

Divide each term in <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi><mo>=</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>7</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Cancel the common factor of <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> .

Cancel the common factor.

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

Move the negative in front of the fraction.

Rewrite the equation in vertex form.

Complete the square for <math><mstyle displaystyle="true"><mfrac><mrow><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><mn>3</mn></mrow></mfrac><mo>-</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>3</mn></mrow></mfrac></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> .

Simplify the right side.

Cancel the common factor of <math><mstyle displaystyle="true"><mn>0</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>0</mn></mstyle></math> .

Cancel the common factors.

Cancel the common factor.

Rewrite the expression.

Multiply the numerator by the reciprocal of the denominator.

Multiply <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>3</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.

Raising <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> to any positive power yields <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

Multiply the numerator by the reciprocal of the denominator.

Multiply <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> by <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> and <math><mstyle displaystyle="true"><mn>0</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>x</mi></mstyle></math> equal to the new right side.

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

Opens Right

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.

Simplify.

Combine <math><mstyle displaystyle="true"><mn>4</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>1</mn></mrow><mrow><mn>3</mn></mrow></mfrac></mstyle></math> .

Multiply the numerator by the reciprocal of the denominator.

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Find the focus.

The focus of a parabola can be found by adding <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> to the x-coordinate <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> if the parabola opens left or right.

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 vertical line found by subtracting <math><mstyle displaystyle="true"><mi>p</mi></mstyle></math> from the x-coordinate <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> of the vertex if the parabola opens left or right.

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

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

Direction: Opens Right

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mn>19</mn></mrow><mrow><mn>12</mn></mrow></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>0</mn></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>37</mn></mrow><mrow><mn>12</mn></mrow></mfrac></mstyle></math>

Direction: Opens Right

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mn>19</mn></mrow><mrow><mn>12</mn></mrow></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>0</mn></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>37</mn></mrow><mrow><mn>12</mn></mrow></mfrac></mstyle></math>

Substitute the <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> value <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>1.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> into <math><mstyle displaystyle="true"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></msqrt></mstyle></math> . In this case, the point is <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn><mn>1.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn><mo>,</mo><msqrt><mn>3</mn></msqrt><mo>)</mo></mrow></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>1.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> in the expression.

Simplify the result.

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>1.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>4</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> .

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

Substitute the <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> value <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>1.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> into <math><mstyle displaystyle="true"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>-</mo><msqrt><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></msqrt></mstyle></math> . In this case, the point is <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn><mn>1.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn><mo>,</mo><mo>-</mo><msqrt><mn>3</mn></msqrt><mo>)</mo></mrow></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>1.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> in the expression.

Simplify the result.

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>1.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>4</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> .

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

Substitute the <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> value <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> into <math><mstyle displaystyle="true"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msqrt><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></msqrt></mstyle></math> . In this case, the point is <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn><mo>,</mo><msqrt><mn>6</mn></msqrt><mo>)</mo></mrow></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> in the expression.

Simplify the result.

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><msqrt><mn>6</mn></msqrt></mstyle></math> .

Substitute the <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> value <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> into <math><mstyle displaystyle="true"><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><mo>-</mo><msqrt><mn>3</mn><mi>x</mi><mo>+</mo><mn>7</mn></msqrt></mstyle></math> . In this case, the point is <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn><mo>,</mo><mo>-</mo><msqrt><mn>6</mn></msqrt><mo>)</mo></mrow></mstyle></math> .

Replace the variable <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> with <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> in the expression.

Simplify the result.

Multiply <math><mstyle displaystyle="true"><mn>3</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn><mn>0.</mn><mover accent="true"><mn>3</mn><mo>‾</mo></mover></mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>7</mn></mstyle></math> .

The final answer is <math><mstyle displaystyle="true"><mo>-</mo><msqrt><mn>6</mn></msqrt></mstyle></math> .

Graph the parabola using its properties and the selected points.

Graph the parabola using its properties and the selected points.

Direction: Opens Right

Vertex: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math>

Focus: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mo>-</mo><mfrac><mrow><mn>19</mn></mrow><mrow><mn>12</mn></mrow></mfrac><mo>,</mo><mn>0</mn><mo>)</mo></mrow></mstyle></math>

Axis of Symmetry: <math><mstyle displaystyle="true"><mi>y</mi><mo>=</mo><mn>0</mn></mstyle></math>

Directrix: <math><mstyle displaystyle="true"><mi>x</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>37</mn></mrow><mrow><mn>12</mn></mrow></mfrac></mstyle></math>

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Name | five hundred eighty-one million three hundred seventy-five thousand nine hundred forty-two |
---|

- 581375942 has 32 divisors, whose sum is
**1013030400** - The reverse of 581375942 is
**249573185** - Previous prime number is
**73**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 44
- Digital Root 8

Name | one hundred six million one hundred seventy-three thousand four hundred eleven |
---|

- 106173411 has 8 divisors, whose sum is
**141705984** - The reverse of 106173411 is
**114371601** - Previous prime number is
**1031**

- Is Prime? no
- Number parity odd
- Number length 9
- Sum of Digits 24
- Digital Root 6

Name | five hundred four million five hundred seventeen thousand seven hundred six |
---|

- 504517706 has 32 divisors, whose sum is
**961331328** - The reverse of 504517706 is
**607715405** - Previous prime number is
**53**

- Is Prime? no
- Number parity even
- Number length 9
- Sum of Digits 35
- Digital Root 8