Rewrite the equation in term of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mi>y</mi></mstyle></math> .

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

Simplify the right side.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> .

Move the negative in front of the fraction.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mo>-</mo><mfrac><mrow><mn>7</mn></mrow><mrow><mn>6</mn></mrow></mfrac></mstyle></math> .

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

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>7</mn></mrow><mrow><mn>6</mn></mrow></mfrac></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"><mo>-</mo><mn>7</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"><mo>-</mo><mn>3</mn></mstyle></math> .

Move the negative in front of the fraction.

Multiply <math><mstyle displaystyle="true"><mo>-</mo><mo>-</mo><mfrac><mrow><mn>49</mn></mrow><mrow><mn>12</mn></mrow></mfrac></mstyle></math> .

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

Multiply <math><mstyle displaystyle="true"><mfrac><mrow><mn>49</mn></mrow><mrow><mn>12</mn></mrow></mfrac></mstyle></math> by <math><mstyle displaystyle="true"><mn>1</mn></mstyle></math> .

To write <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> as a fraction with a common denominator, multiply by <math><mstyle displaystyle="true"><mfrac><mrow><mn>12</mn></mrow><mrow><mn>12</mn></mrow></mfrac></mstyle></math> .

Combine <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> and <math><mstyle displaystyle="true"><mfrac><mrow><mn>12</mn></mrow><mrow><mn>12</mn></mrow></mfrac></mstyle></math> .

Combine the numerators over the common denominator.

Simplify the numerator.

Multiply <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> by <math><mstyle displaystyle="true"><mn>12</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>24</mn></mstyle></math> and <math><mstyle displaystyle="true"><mn>49</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> .

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

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