Finite Math Examples

Solve for x (x-2)/( square root of 4x^2-19x+30)=1/3
Step 1
Cross multiply.
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Step 1.1
Cross multiply by setting the product of the numerator of the right side and the denominator of the left side equal to the product of the numerator of the left side and the denominator of the right side.
Step 1.2
Simplify the left side.
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Step 1.2.1
Multiply by .
Step 1.3
Simplify the right side.
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Step 1.3.1
Simplify .
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Step 1.3.1.1
Apply the distributive property.
Step 1.3.1.2
Simplify the expression.
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Step 1.3.1.2.1
Move to the left of .
Step 1.3.1.2.2
Multiply by .
Step 2
To remove the radical on the left side of the equation, square both sides of the equation.
Step 3
Simplify each side of the equation.
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Step 3.1
Use to rewrite as .
Step 3.2
Simplify the left side.
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Step 3.2.1
Simplify .
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Step 3.2.1.1
Multiply the exponents in .
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Step 3.2.1.1.1
Apply the power rule and multiply exponents, .
Step 3.2.1.1.2
Cancel the common factor of .
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Step 3.2.1.1.2.1
Cancel the common factor.
Step 3.2.1.1.2.2
Rewrite the expression.
Step 3.2.1.2
Simplify.
Step 3.3
Simplify the right side.
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Step 3.3.1
Simplify .
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Step 3.3.1.1
Rewrite as .
Step 3.3.1.2
Expand using the FOIL Method.
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Step 3.3.1.2.1
Apply the distributive property.
Step 3.3.1.2.2
Apply the distributive property.
Step 3.3.1.2.3
Apply the distributive property.
Step 3.3.1.3
Simplify and combine like terms.
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Step 3.3.1.3.1
Simplify each term.
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Step 3.3.1.3.1.1
Rewrite using the commutative property of multiplication.
Step 3.3.1.3.1.2
Multiply by by adding the exponents.
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Step 3.3.1.3.1.2.1
Move .
Step 3.3.1.3.1.2.2
Multiply by .
Step 3.3.1.3.1.3
Multiply by .
Step 3.3.1.3.1.4
Multiply by .
Step 3.3.1.3.1.5
Multiply by .
Step 3.3.1.3.1.6
Multiply by .
Step 3.3.1.3.2
Subtract from .
Step 4
Solve for .
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Step 4.1
Move all terms containing to the left side of the equation.
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Step 4.1.1
Subtract from both sides of the equation.
Step 4.1.2
Add to both sides of the equation.
Step 4.1.3
Subtract from .
Step 4.1.4
Add and .
Step 4.2
Subtract from both sides of the equation.
Step 4.3
Subtract from .
Step 4.4
Factor the left side of the equation.
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Step 4.4.1
Factor out of .
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Step 4.4.1.1
Factor out of .
Step 4.4.1.2
Factor out of .
Step 4.4.1.3
Rewrite as .
Step 4.4.1.4
Factor out of .
Step 4.4.1.5
Factor out of .
Step 4.4.2
Factor.
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Step 4.4.2.1
Factor by grouping.
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Step 4.4.2.1.1
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 4.4.2.1.1.1
Factor out of .
Step 4.4.2.1.1.2
Rewrite as plus
Step 4.4.2.1.1.3
Apply the distributive property.
Step 4.4.2.1.2
Factor out the greatest common factor from each group.
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Step 4.4.2.1.2.1
Group the first two terms and the last two terms.
Step 4.4.2.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 4.4.2.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 4.4.2.2
Remove unnecessary parentheses.
Step 4.5
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 4.6
Set equal to and solve for .
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Step 4.6.1
Set equal to .
Step 4.6.2
Solve for .
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Step 4.6.2.1
Add to both sides of the equation.
Step 4.6.2.2
Divide each term in by and simplify.
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Step 4.6.2.2.1
Divide each term in by .
Step 4.6.2.2.2
Simplify the left side.
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Step 4.6.2.2.2.1
Cancel the common factor of .
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Step 4.6.2.2.2.1.1
Cancel the common factor.
Step 4.6.2.2.2.1.2
Divide by .
Step 4.7
Set equal to and solve for .
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Step 4.7.1
Set equal to .
Step 4.7.2
Add to both sides of the equation.
Step 4.8
The final solution is all the values that make true.
Step 5
Exclude the solutions that do not make true.