Linear Algebra Examples

Find the Domain p^2+2q^2=11
Step 1
Subtract from both sides of the equation.
Step 2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 3
The complete solution is the result of both the positive and negative portions of the solution.
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Step 3.1
First, use the positive value of the to find the first solution.
Step 3.2
Next, use the negative value of the to find the second solution.
Step 3.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 4
Set the radicand in greater than or equal to to find where the expression is defined.
Step 5
Solve for .
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Step 5.1
Subtract from both sides of the inequality.
Step 5.2
Divide each term in by and simplify.
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Step 5.2.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 5.2.2
Simplify the left side.
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Step 5.2.2.1
Cancel the common factor of .
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Step 5.2.2.1.1
Cancel the common factor.
Step 5.2.2.1.2
Divide by .
Step 5.2.3
Simplify the right side.
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Step 5.2.3.1
Dividing two negative values results in a positive value.
Step 5.3
Take the specified root of both sides of the inequality to eliminate the exponent on the left side.
Step 5.4
Simplify the equation.
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Step 5.4.1
Simplify the left side.
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Step 5.4.1.1
Pull terms out from under the radical.
Step 5.4.2
Simplify the right side.
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Step 5.4.2.1
Simplify .
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Step 5.4.2.1.1
Rewrite as .
Step 5.4.2.1.2
Multiply by .
Step 5.4.2.1.3
Combine and simplify the denominator.
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Step 5.4.2.1.3.1
Multiply by .
Step 5.4.2.1.3.2
Raise to the power of .
Step 5.4.2.1.3.3
Raise to the power of .
Step 5.4.2.1.3.4
Use the power rule to combine exponents.
Step 5.4.2.1.3.5
Add and .
Step 5.4.2.1.3.6
Rewrite as .
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Step 5.4.2.1.3.6.1
Use to rewrite as .
Step 5.4.2.1.3.6.2
Apply the power rule and multiply exponents, .
Step 5.4.2.1.3.6.3
Combine and .
Step 5.4.2.1.3.6.4
Cancel the common factor of .
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Step 5.4.2.1.3.6.4.1
Cancel the common factor.
Step 5.4.2.1.3.6.4.2
Rewrite the expression.
Step 5.4.2.1.3.6.5
Evaluate the exponent.
Step 5.4.2.1.4
Simplify the numerator.
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Step 5.4.2.1.4.1
Combine using the product rule for radicals.
Step 5.4.2.1.4.2
Multiply by .
Step 5.5
Write as a piecewise.
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Step 5.5.1
To find the interval for the first piece, find where the inside of the absolute value is non-negative.
Step 5.5.2
In the piece where is non-negative, remove the absolute value.
Step 5.5.3
To find the interval for the second piece, find where the inside of the absolute value is negative.
Step 5.5.4
In the piece where is negative, remove the absolute value and multiply by .
Step 5.5.5
Write as a piecewise.
Step 5.6
Find the intersection of and .
Step 5.7
Solve when .
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Step 5.7.1
Divide each term in by and simplify.
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Step 5.7.1.1
Divide each term in by . When multiplying or dividing both sides of an inequality by a negative value, flip the direction of the inequality sign.
Step 5.7.1.2
Simplify the left side.
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Step 5.7.1.2.1
Dividing two negative values results in a positive value.
Step 5.7.1.2.2
Divide by .
Step 5.7.1.3
Simplify the right side.
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Step 5.7.1.3.1
Move the negative one from the denominator of .
Step 5.7.1.3.2
Rewrite as .
Step 5.7.2
Find the intersection of and .
Step 5.8
Find the union of the solutions.
Step 6
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 7