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Finite Math Examples
Step 1
Use the quadratic formula to find the solutions.
Step 2
Substitute the values , , and into the quadratic formula and solve for .
Step 3
Step 3.1
Simplify the numerator.
Step 3.1.1
Raise to the power of .
Step 3.1.2
Multiply by .
Step 3.1.3
Apply the distributive property.
Step 3.1.4
Simplify.
Step 3.1.4.1
Multiply by .
Step 3.1.4.2
Multiply by .
Step 3.1.5
Subtract from .
Step 3.1.6
Rewrite in a factored form.
Step 3.1.6.1
Factor out of .
Step 3.1.6.1.1
Factor out of .
Step 3.1.6.1.2
Factor out of .
Step 3.1.6.1.3
Factor out of .
Step 3.1.6.1.4
Factor out of .
Step 3.1.6.1.5
Factor out of .
Step 3.1.6.2
Factor by grouping.
Step 3.1.6.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 3.1.6.2.1.1
Factor out of .
Step 3.1.6.2.1.2
Rewrite as plus
Step 3.1.6.2.1.3
Apply the distributive property.
Step 3.1.6.2.2
Factor out the greatest common factor from each group.
Step 3.1.6.2.2.1
Group the first two terms and the last two terms.
Step 3.1.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 3.1.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 3.1.7
Rewrite as .
Step 3.1.7.1
Rewrite as .
Step 3.1.7.2
Add parentheses.
Step 3.1.8
Pull terms out from under the radical.
Step 3.2
Multiply by .
Step 3.3
Simplify .
Step 4
Step 4.1
Simplify the numerator.
Step 4.1.1
Raise to the power of .
Step 4.1.2
Multiply by .
Step 4.1.3
Apply the distributive property.
Step 4.1.4
Simplify.
Step 4.1.4.1
Multiply by .
Step 4.1.4.2
Multiply by .
Step 4.1.5
Subtract from .
Step 4.1.6
Rewrite in a factored form.
Step 4.1.6.1
Factor out of .
Step 4.1.6.1.1
Factor out of .
Step 4.1.6.1.2
Factor out of .
Step 4.1.6.1.3
Factor out of .
Step 4.1.6.1.4
Factor out of .
Step 4.1.6.1.5
Factor out of .
Step 4.1.6.2
Factor by grouping.
Step 4.1.6.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 4.1.6.2.1.1
Factor out of .
Step 4.1.6.2.1.2
Rewrite as plus
Step 4.1.6.2.1.3
Apply the distributive property.
Step 4.1.6.2.2
Factor out the greatest common factor from each group.
Step 4.1.6.2.2.1
Group the first two terms and the last two terms.
Step 4.1.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 4.1.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 4.1.7
Rewrite as .
Step 4.1.7.1
Rewrite as .
Step 4.1.7.2
Add parentheses.
Step 4.1.8
Pull terms out from under the radical.
Step 4.2
Multiply by .
Step 4.3
Simplify .
Step 4.4
Change the to .
Step 5
Step 5.1
Simplify the numerator.
Step 5.1.1
Raise to the power of .
Step 5.1.2
Multiply by .
Step 5.1.3
Apply the distributive property.
Step 5.1.4
Simplify.
Step 5.1.4.1
Multiply by .
Step 5.1.4.2
Multiply by .
Step 5.1.5
Subtract from .
Step 5.1.6
Rewrite in a factored form.
Step 5.1.6.1
Factor out of .
Step 5.1.6.1.1
Factor out of .
Step 5.1.6.1.2
Factor out of .
Step 5.1.6.1.3
Factor out of .
Step 5.1.6.1.4
Factor out of .
Step 5.1.6.1.5
Factor out of .
Step 5.1.6.2
Factor by grouping.
Step 5.1.6.2.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
Step 5.1.6.2.1.1
Factor out of .
Step 5.1.6.2.1.2
Rewrite as plus
Step 5.1.6.2.1.3
Apply the distributive property.
Step 5.1.6.2.2
Factor out the greatest common factor from each group.
Step 5.1.6.2.2.1
Group the first two terms and the last two terms.
Step 5.1.6.2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 5.1.6.2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 5.1.7
Rewrite as .
Step 5.1.7.1
Rewrite as .
Step 5.1.7.2
Add parentheses.
Step 5.1.8
Pull terms out from under the radical.
Step 5.2
Multiply by .
Step 5.3
Simplify .
Step 5.4
Change the to .
Step 6
The final answer is the combination of both solutions.
Step 7
Set the radicand in greater than or equal to to find where the expression is defined.
Step 8
Step 8.1
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 8.2
Set equal to and solve for .
Step 8.2.1
Set equal to .
Step 8.2.2
Solve for .
Step 8.2.2.1
Add to both sides of the equation.
Step 8.2.2.2
Divide each term in by and simplify.
Step 8.2.2.2.1
Divide each term in by .
Step 8.2.2.2.2
Simplify the left side.
Step 8.2.2.2.2.1
Dividing two negative values results in a positive value.
Step 8.2.2.2.2.2
Divide by .
Step 8.2.2.2.3
Simplify the right side.
Step 8.2.2.2.3.1
Divide by .
Step 8.3
Set equal to and solve for .
Step 8.3.1
Set equal to .
Step 8.3.2
Subtract from both sides of the equation.
Step 8.4
The final solution is all the values that make true.
Step 8.5
Use each root to create test intervals.
Step 8.6
Choose a test value from each interval and plug this value into the original inequality to determine which intervals satisfy the inequality.
Step 8.6.1
Test a value on the interval to see if it makes the inequality true.
Step 8.6.1.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 8.6.1.2
Replace with in the original inequality.
Step 8.6.1.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 8.6.2
Test a value on the interval to see if it makes the inequality true.
Step 8.6.2.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 8.6.2.2
Replace with in the original inequality.
Step 8.6.2.3
The left side is greater than the right side , which means that the given statement is always true.
True
True
Step 8.6.3
Test a value on the interval to see if it makes the inequality true.
Step 8.6.3.1
Choose a value on the interval and see if this value makes the original inequality true.
Step 8.6.3.2
Replace with in the original inequality.
Step 8.6.3.3
The left side is less than the right side , which means that the given statement is false.
False
False
Step 8.6.4
Compare the intervals to determine which ones satisfy the original inequality.
False
True
False
False
True
False
Step 8.7
The solution consists of all of the true intervals.
Step 9
The domain is all values of that make the expression defined.
Interval Notation:
Set-Builder Notation:
Step 10
The range is the set of all valid values. Use the graph to find the range.
Interval Notation:
Set-Builder Notation:
Step 11
Determine the domain and range.
Domain:
Range:
Step 12