Linear Algebra Examples

Find the Domain x=(-b+ square root of b^2-4ac)/(2a)
Step 1
Simplify .
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Step 1.1
Factor out of .
Step 1.2
Factor out of .
Step 1.3
Factor out of .
Step 1.4
Simplify the expression.
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Step 1.4.1
Rewrite as .
Step 1.4.2
Move the negative in front of the fraction.
Step 2
Set the radicand in greater than or equal to to find where the expression is defined.
Step 3
Solve for .
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Step 3.1
Add to both sides of the inequality.
Step 3.2
Take the specified root of both sides of the inequality to eliminate the exponent on the left side.
Step 3.3
Simplify the equation.
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Step 3.3.1
Simplify the left side.
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Step 3.3.1.1
Pull terms out from under the radical.
Step 3.3.2
Simplify the right side.
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Step 3.3.2.1
Simplify .
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Step 3.3.2.1.1
Rewrite as .
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Step 3.3.2.1.1.1
Rewrite as .
Step 3.3.2.1.1.2
Add parentheses.
Step 3.3.2.1.2
Pull terms out from under the radical.
Step 3.3.2.1.3
The absolute value is the distance between a number and zero. The distance between and is .
Step 3.4
Write as a piecewise.
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Step 3.4.1
To find the interval for the first piece, find where the inside of the absolute value is non-negative.
Step 3.4.2
In the piece where is non-negative, remove the absolute value.
Step 3.4.3
Find the domain of and find the intersection with .
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Step 3.4.3.1
Find the domain of .
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Step 3.4.3.1.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 3.4.3.1.2
Divide each term in by and simplify.
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Step 3.4.3.1.2.1
Divide each term in by .
Step 3.4.3.1.2.2
Simplify the left side.
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Step 3.4.3.1.2.2.1
Cancel the common factor of .
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Step 3.4.3.1.2.2.1.1
Cancel the common factor.
Step 3.4.3.1.2.2.1.2
Divide by .
Step 3.4.3.1.2.3
Simplify the right side.
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Step 3.4.3.1.2.3.1
Divide by .
Step 3.4.3.1.3
The domain is all values of that make the expression defined.
Step 3.4.3.2
Find the intersection of and .
Step 3.4.4
To find the interval for the second piece, find where the inside of the absolute value is negative.
Step 3.4.5
In the piece where is negative, remove the absolute value and multiply by .
Step 3.4.6
Find the domain of and find the intersection with .
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Step 3.4.6.1
Find the domain of .
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Step 3.4.6.1.1
Set the radicand in greater than or equal to to find where the expression is defined.
Step 3.4.6.1.2
Divide each term in by and simplify.
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Step 3.4.6.1.2.1
Divide each term in by .
Step 3.4.6.1.2.2
Simplify the left side.
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Step 3.4.6.1.2.2.1
Cancel the common factor of .
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Step 3.4.6.1.2.2.1.1
Cancel the common factor.
Step 3.4.6.1.2.2.1.2
Divide by .
Step 3.4.6.1.2.3
Simplify the right side.
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Step 3.4.6.1.2.3.1
Divide by .
Step 3.4.6.1.3
The domain is all values of that make the expression defined.
Step 3.4.6.2
Find the intersection of and .
Step 3.4.7
Write as a piecewise.
Step 3.5
Find the intersection of and .
and
Step 3.6
Find the union of the solutions.
Step 4
Set the denominator in equal to to find where the expression is undefined.
Step 5
Divide each term in by and simplify.
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Step 5.1
Divide each term in by .
Step 5.2
Simplify the left side.
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Step 5.2.1
Cancel the common factor of .
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Step 5.2.1.1
Cancel the common factor.
Step 5.2.1.2
Divide by .
Step 5.3
Simplify the right side.
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Step 5.3.1
Divide by .
Step 6
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 7