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>10</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>10</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>5</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mn>1</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>5</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"><mo>-</mo><mn>10</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>100</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>25</mn></mstyle></math> .

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

Substitute <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>5</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>25</mn></mstyle></math> for <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>10</mn><mi>x</mi></mstyle></math> in the equation <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>18</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>42</mn></mstyle></math> .

Move <math><mstyle displaystyle="true"><mo>-</mo><mn>25</mn></mstyle></math> to the right side of the equation by adding <math><mstyle displaystyle="true"><mn>25</mn></mstyle></math> to both sides.

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>18</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>18</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>9</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>18</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>324</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>81</mn></mstyle></math> .

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

Substitute <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>y</mi><mo>+</mo><mn>9</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>81</mn></mstyle></math> for <math><mstyle displaystyle="true"><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mn>18</mn><mi>y</mi></mstyle></math> in the equation <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msup><mrow><mi>y</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>18</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>42</mn></mstyle></math> .

Move <math><mstyle displaystyle="true"><mo>-</mo><mn>81</mn></mstyle></math> to the right side of the equation by adding <math><mstyle displaystyle="true"><mn>81</mn></mstyle></math> to both sides.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>42</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>25</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>17</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>81</mn></mstyle></math> .

This is the form of a circle. Use this form to determine the center and radius of the circle.

Match the values in this circle to those of the standard form. The variable <math><mstyle displaystyle="true"><mi>r</mi></mstyle></math> represents the radius of the circle, <math><mstyle displaystyle="true"><mi>h</mi></mstyle></math> represents the x-offset from the origin, and <math><mstyle displaystyle="true"><mi>k</mi></mstyle></math> represents the y-offset from origin.

The center of the circle is found at <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>h</mi><mo>,</mo><mi>k</mi><mo>)</mo></mrow></mstyle></math> .

Center: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>5</mn><mo>,</mo><mo>-</mo><mn>9</mn><mo>)</mo></mrow></mstyle></math>

These values represent the important values for graphing and analyzing a circle.

Center: <math><mstyle displaystyle="true"><mrow><mo>(</mo><mn>5</mn><mo>,</mo><mo>-</mo><mn>9</mn><mo>)</mo></mrow></mstyle></math>

Radius: <math><mstyle displaystyle="true"><mn>8</mn></mstyle></math>

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