Basic Math Examples

Solve for z 5z^4+16z^2-45=0
Step 1
Substitute into the equation. This will make the quadratic formula easy to use.
Step 2
Factor by grouping.
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Step 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 .
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Step 2.1.1
Factor out of .
Step 2.1.2
Rewrite as plus
Step 2.1.3
Apply the distributive property.
Step 2.2
Factor out the greatest common factor from each group.
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Step 2.2.1
Group the first two terms and the last two terms.
Step 2.2.2
Factor out the greatest common factor (GCF) from each group.
Step 2.3
Factor the polynomial by factoring out the greatest common factor, .
Step 3
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4
Set equal to and solve for .
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Step 4.1
Set equal to .
Step 4.2
Solve for .
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Step 4.2.1
Add to both sides of the equation.
Step 4.2.2
Divide each term in by and simplify.
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Step 4.2.2.1
Divide each term in by .
Step 4.2.2.2
Simplify the left side.
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Step 4.2.2.2.1
Cancel the common factor of .
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Step 4.2.2.2.1.1
Cancel the common factor.
Step 4.2.2.2.1.2
Divide by .
Step 5
Set equal to and solve for .
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Step 5.1
Set equal to .
Step 5.2
Subtract from both sides of the equation.
Step 6
The final solution is all the values that make true.
Step 7
Substitute the real value of back into the solved equation.
Step 8
Solve the first equation for .
Step 9
Solve the equation for .
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Step 9.1
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 9.2
Simplify .
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Step 9.2.1
Rewrite as .
Step 9.2.2
Simplify the numerator.
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Step 9.2.2.1
Rewrite as .
Step 9.2.2.2
Pull terms out from under the radical, assuming positive real numbers.
Step 9.2.3
Multiply by .
Step 9.2.4
Combine and simplify the denominator.
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Step 9.2.4.1
Multiply by .
Step 9.2.4.2
Raise to the power of .
Step 9.2.4.3
Raise to the power of .
Step 9.2.4.4
Use the power rule to combine exponents.
Step 9.2.4.5
Add and .
Step 9.2.4.6
Rewrite as .
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Step 9.2.4.6.1
Use to rewrite as .
Step 9.2.4.6.2
Apply the power rule and multiply exponents, .
Step 9.2.4.6.3
Combine and .
Step 9.2.4.6.4
Cancel the common factor of .
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Step 9.2.4.6.4.1
Cancel the common factor.
Step 9.2.4.6.4.2
Rewrite the expression.
Step 9.2.4.6.5
Evaluate the exponent.
Step 9.3
The complete solution is the result of both the positive and negative portions of the solution.
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Step 9.3.1
First, use the positive value of the to find the first solution.
Step 9.3.2
Next, use the negative value of the to find the second solution.
Step 9.3.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 10
Solve the second equation for .
Step 11
Solve the equation for .
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Step 11.1
Remove parentheses.
Step 11.2
Take the specified root of both sides of the equation to eliminate the exponent on the left side.
Step 11.3
Simplify .
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Step 11.3.1
Rewrite as .
Step 11.3.2
Rewrite as .
Step 11.3.3
Rewrite as .
Step 11.4
The complete solution is the result of both the positive and negative portions of the solution.
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Step 11.4.1
First, use the positive value of the to find the first solution.
Step 11.4.2
Next, use the negative value of the to find the second solution.
Step 11.4.3
The complete solution is the result of both the positive and negative portions of the solution.
Step 12
The solution to is .