Since the radical is on the right side of the equation, switch the sides so it is on the left side of the equation.

To remove the radical on the left side of the equation, square both sides of the equation.

Use <math><mstyle displaystyle="true"><mroot><mrow><msup><mrow><mi>a</mi></mrow><mrow><mi>x</mi></mrow></msup></mrow><mrow><mi>n</mi></mrow></mroot><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mfrac><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></mfrac></mrow></msup></mstyle></math> to rewrite <math><mstyle displaystyle="true"><msqrt><mn>19</mn><mo>-</mo><mn>3</mn><mi>x</mi></msqrt></mstyle></math> as <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mn>19</mn><mo>-</mo><mn>3</mn><mi>x</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup></mstyle></math> .

Simplify the left side.

Simplify <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mo>(</mo><mn>19</mn><mo>-</mo><mn>3</mn><mi>x</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Multiply the exponents in <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mo>(</mo><mn>19</mn><mo>-</mo><mn>3</mn><mi>x</mi><mo>)</mo></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac></mrow></msup><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Apply the power rule and multiply exponents, <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi></mrow></msup><mo>)</mo></mrow><mrow><mi>n</mi></mrow></msup><mo>=</mo><msup><mrow><mi>a</mi></mrow><mrow><mi>m</mi><mi>n</mi></mrow></msup></mstyle></math> .

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

Cancel the common factor.

Rewrite the expression.

Simplify.

Simplify the right side.

Simplify <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> .

Rewrite <math><mstyle displaystyle="true"><msup><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></msup></mstyle></math> as <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow></mstyle></math> .

Expand <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>3</mn><mo>)</mo></mrow></mstyle></math> using the FOIL Method.

Apply the distributive property.

Apply the distributive property.

Apply the distributive property.

Simplify and combine like terms.

Simplify each term.

Multiply <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> by <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Move <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> to the left of <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

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

Subtract <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi></mstyle></math> from <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn><mi>x</mi></mstyle></math> .

Since <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> is on the right side of the equation, switch the sides so it is on the left side of the equation.

Move all terms containing <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> to the left side of the equation.

Add <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi></mstyle></math> to both sides of the equation.

Add <math><mstyle displaystyle="true"><mo>-</mo><mn>6</mn><mi>x</mi></mstyle></math> and <math><mstyle displaystyle="true"><mn>3</mn><mi>x</mi></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>19</mn></mstyle></math> from both sides of the equation.

Subtract <math><mstyle displaystyle="true"><mn>19</mn></mstyle></math> from <math><mstyle displaystyle="true"><mn>9</mn></mstyle></math> .

Factor <math><mstyle displaystyle="true"><msup><mrow><mi>x</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>-</mo><mn>3</mn><mi>x</mi><mo>-</mo><mn>10</mn></mstyle></math> using the AC method.

Consider the form <math><mstyle displaystyle="true"><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> . Find a pair of integers whose product is <math><mstyle displaystyle="true"><mi>c</mi></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mi>b</mi></mstyle></math> . In this case, whose product is <math><mstyle displaystyle="true"><mo>-</mo><mn>10</mn></mstyle></math> and whose sum is <math><mstyle displaystyle="true"><mo>-</mo><mn>3</mn></mstyle></math> .

Write the factored form using these integers.

If any individual factor on the left side of the equation is equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> , the entire expression will be equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Set <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mn>5</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Set <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mn>5</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Add <math><mstyle displaystyle="true"><mn>5</mn></mstyle></math> to both sides of the equation.

Set <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>2</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> and solve for <math><mstyle displaystyle="true"><mi>x</mi></mstyle></math> .

Set <math><mstyle displaystyle="true"><mi>x</mi><mo>+</mo><mn>2</mn></mstyle></math> equal to <math><mstyle displaystyle="true"><mn>0</mn></mstyle></math> .

Subtract <math><mstyle displaystyle="true"><mn>2</mn></mstyle></math> from both sides of the equation.

The final solution is all the values that make <math><mstyle displaystyle="true"><mrow><mo>(</mo><mi>x</mi><mo>-</mo><mn>5</mn><mo>)</mo></mrow><mrow><mo>(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mo>)</mo></mrow><mo>=</mo><mn>0</mn></mstyle></math> true.

Exclude the solutions that do not make <math><mstyle displaystyle="true"><mi>x</mi><mo>-</mo><mn>3</mn><mo>=</mo><msqrt><mn>19</mn><mo>-</mo><mn>3</mn><mi>x</mi></msqrt></mstyle></math> true.

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