All GMAT Math Resources
Example Questions
Example Question #11 : Solving By Factoring
Factor the expression completely:
The expression is a perfect square trinomial, as the three terms have the following relationship:
We can factor this expression by substituting into the following pattern:
We can factor further by noting that , the difference of squares, and subsequently, factoring this as the product of a sum and a difference.
or
Example Question #11 : Solving By Factoring
If what is ?
Note that
Therefore is equivalent to:
Example Question #422 : Algebra
Solve for ;
None of the other answers
None of the other answers
The correct answer is . Our work proceeds as follows:
(Factor an out of the numerator and denominator)
(Factor the quadratic polynomials)
(Cancel common terms)
(Multiply by to both sides)
(Distribute the )
(Simplify and solve)
Example Question #1504 : Problem Solving Questions
Solve for by factoring and using the zero product property.
In order to solve for we must first factor:
The zero product property states that if then or (or both).
Our two equations are then:
Solving for in each leaves us with:
and
Example Question #1505 : Problem Solving Questions
Find the roots of the following function:
The roots of a function are the points at which it crosses the x axis, so at these points the value of y, or f(x), is 0. This gives us:
So we will have to factor the polynomial in order to solve for the x values at which the function is equal to 0. We need two factors whose product is -18 and whose sum is -3. If we think about our options, 2 and 9 have a product of -18 if one is negative, but there's no way of making these two numbers add up to -3. Next we consider 3 and 6. These numbers have a product of -18 if one is negative, and their sum can also be -3 if the 3 is positive and the 6 is negative. This allows us to write out the following factorization:
Example Question #1506 : Problem Solving Questions
Where does the following function cross the -axis?
We could solve this question a variety of ways. The simplest would be graphing with a calculator, but we will use factoring.
To begin, set our function equal to . We want to find where this function crosses the -axis—in other words, where .
Next, we need to factor the function into two binomial terms. Remember FOIL/box method? We are essentially doing the reverse here. We are looking for something in the form of .
Recalling a few details will make this easier.
1) must equal positive
2) and must both be negative, because we get positive when we multiply them and when we add them.
3) and must be factors of that add up to . List factors of : . The only pair of those that will add up to are and , so our factored form looks like this:
Then, due to the zero product property, we know that if or one side of the equation will equal , and therefore our answers are positive and positive .
Example Question #21 : Solving By Factoring
Which of the following is the set of possible values of ?
If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:
Either or
Similarly,
Either or
Four systems of equations can be set up here, and in each case, we can find by adding both sides:
The set of possible values of is .
Example Question #21 : Solving By Factoring
Which of the following is the set of possible values of ?
If you note that the expressions on the left sides of the equations are perfect square trinomials, you can rewrite the expressions as follows:
Either or
Similarly,
Either or
Four systems of equations can be set up here, and in each case, we can find by subtracting both sides:
The set of possible values of is .
Example Question #21 : Solving By Factoring
Solve for by factoring:
When we factor a polynomial, we are left with two factors of the form:
For the given function, this means we must find two factors and whose product is 10 and whose sum is 7. Thinking about factors of 10, we can see that 1 and 10 cannot add up to 7 in any way. The only two factors left, then, are 2 and 5, which we can see have a product of 10 and a sum of 7. This gives us:
Example Question #21 : Solving By Factoring
Find the roots of the following function:
The roots of a function are the points at which it crosses the axis, so will be equal to zero at these points. This means we set the function equal to zero, and then factor it to solve for the values of the roots:
Now that we've factored out a , we can see we need two factors whose product is and whose sum is . Thinking about the possible factors, we can see that and have a product of and a sum of . This gives us: