All GMAT Math Resources
Example Questions
Example Question #2 : Solving Quadratic Equations
Define an operation as follows:
For all real numbers ,
Solve for :
Subsitute in the defintion, and set it equal to 21:
This sets up a quadratic equation, Move all terms to the left, factor the expression, set each factor to 0, and solve separately.
or
The solution set is
Example Question #4 : Solving Quadratic Equations
Solve for ;
First, rewrite the quadratic equation in standard form by FOILing out the product on the left, then colleciting all of the terms on the left side:
Now factor the quadratic expression to two binomial factors , replacing the question marks with two integers whose product is 36 and whose sum is . These numbers are , so:
or
The solution set is
Example Question #1 : Solving Quadratic Equations
Solve for :
First, rewrite the quadratic equation in standard form by moving all nonzero terms to the left:
Now factor the quadratic expression into two binomial factors , replacing the question marks with two integers whose product is 16 and whose sum is . These numbers are , so:
or
The solution set is .
Example Question #6 : Solving Quadratic Equations
Find the roots of Separate the answers with a comma.
This can be found by factoring the equation. Doing this we get
We can solve this equation happen when or So the roots are .
Example Question #395 : Algebra
How many real solutions and how many imaginary solutions are there to the following quadratic equation?
No real solutions and one imaginary solution.
Two real solutions and no imaginary solutions.
One real solution and no imaginary solutions.
No real solutions and two imaginary solutions.
One real solution and one imaginary solution.
Two real solutions and no imaginary solutions.
Write the equation in standard form:
Evaluate the discriminant , using .
The discriminant is positive, so the equation has two distinct real solutions.
Example Question #1471 : Problem Solving Questions
How many real solutions and how many imaginary solutions are there to the following equation?
One real solution, no imaginary solutions
One real solution, two imaginary solutions.
No real solutions, three imaginary solutions
No real solutions, two imaginary solutions
Three real solutions, no imaginary solutions
One real solution, two imaginary solutions.
, so this is the difference of cubes.
The difference of cubes can be factored as follows:
Using this pattern and replacing , this becomes:
or
The equation becomes
We set each factor equal to 0and examine the nature of the solutions.
, so the equation has at least one real solution.
is a quadratic equation, so we can explore these two solutions by looking at its discriminant .
We set , so the discriminant is
This negative disciminant indicates that there are two imaginary solutions in addition to the real one we found earlier.
Example Question #1472 : Problem Solving Questions
For what value of does the equation have exactly one solution?
For the equation to have exactly one solution, it must hold that the discriminant , so we solve for this equation setting :
Example Question #1481 : Gmat Quantitative Reasoning
For what value(s) of does this quadratic equation have exactly one solution?
For the equation to have exactly one solution, it must hold that discriminant , so we solve for in this equation, setting :
or
Example Question #1482 : Gmat Quantitative Reasoning
For what value of does this equation have exactly one solution?
For the equation to have exactly one solution, it must hold that discriminant .
Solve for , setting :
Example Question #1483 : Gmat Quantitative Reasoning
For what value(s) of does this equation have exactly one solution?
The discriminant of a quadratic expression is .
For the equation to have exactly one solution, it must hold that discriminant .
Solve for , setting :