Solving Equations

Add nine on both sides.

Divide by negative six on both sides.

The answer is:

can be seen to fit the perfect square trinomial pattern:

The equation can therefore be rewritten as

Multiply both sides of the equation by the least common denominator of the expressions, which is :

This can be solved using the method. We are looking for two integers whose sum is and whose product is . Through some trial and error, the integers are found to be and , so the above equation can be rewritten, and solved using grouping, as

By the Zero Product Principle, one of these factors is equal to zero:

Either:

Or:

Both solutions can be confirmed by substitution; the solution set is .

To isolate the x-variable, we can multiply both sides by the least common denominator.

The least common denominator is . This will eliminate the fractions.

Subtract 4 on both sides.

Divide by 24 on both sides.

The answer is:

First, we want to get everything inside the square roots, so we distribute the :

Now we can clear our the square roots by squaring each side:

Now we can simplify by moving everything to one side of the equation:

Factoring will give us:

So our answers are:

To solve for x, multiply by negative one-third on both sides.

The answer is:

Find the least common denominator of both sides of the equation, and multiply it on both sides.

The LCD is 60.

Combine like-terms on the left.

Divide by 5 on both sides.

The answer is:

First subtract 27 from both sides of the equation

Add 5z to both sides of the equation

Lastly, divide both sides by 5 to get the y by itself

To isolate the x-variable, multiply both sides by the coefficient of the x-variable.

The answer is:

Multiply both sides of the equation by to eliminate the fraction:

Subtract from both sides:

The only possible solution is , However, if this is substituted in the original equation, the expression at left is undefined, as seen here:

An expression with a denominator of 0 has an undefined value, so this statement is false. The equation has no solution.

Begin by gathering all the constants to one side of the equation:

Now multiply by :

And finally, square each side:

This might look all fine and dandy, but let's check our solution by plugging it in to the original equation:

So our solution is invalid, and the problem doesn't have a solution.