# GMAT Math : Solving quadratic equations

## Example Questions

← Previous 1 3

Solve for .

Explanation:

Multiply:

Explanation:

Distribute:

### Example Question #3 : Solving Quadratic Equations

Consider the equation . For what value(s) of  does the equation have two real solutions?

or

or

or

or

Explanation:

The discriminant of the expression  is . For the equation  to have two real solutions, this discriminant must be positive, so:

One of two things happens:

Case 1:

and

and

But this is the same as simply saying

Case 2:

and

and

But this is the same as simply saying

Therefore, the equation has two real solutions if and only if either  or

### Example Question #4 : Solving Quadratic Equations

Find all real solutions for :

The equation has no real solution.

Explanation:

Substitute , and, subsequently, , and solve the resulting quadratic equation.

We can rewrite the quadratic expression as , where the question marks are replaced with integers whose product is  and whose sum is 9; the integers are :

Set each factor to zero and solve for ; then substitute back and solve for :

or

, which has no real solution.

Therefore, the solution set is

### Example Question #5 : Solving Quadratic Equations

What is the minimum value of the function  for all real values of  ?

does not have a minimum value.

Explanation:

We find the -coordinate of the vertex of the parabola for . First, we find its -coordinate using the formula

, setting .

is the -coordindate of the vertex, and, subsequently, the minimum value of :

### Example Question #6 : Solving Quadratic Equations

Define an operation as follows:

For all real numbers ,

Solve for  :

Explanation:

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 #7 : Solving Quadratic Equations

Solve for ;

Explanation:

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 #8 : Solving Quadratic Equations

Solve for :

Explanation:

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 #1 : Solving Quadratic Equations

Find the roots of   Separate the answers with a comma.

Explanation:

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 #10 : Solving Quadratic Equations

How many real solutions and how many imaginary solutions are there to the following quadratic equation?

No real solutions and one imaginary solution.

One real solution and one imaginary solution.

Two real solutions and no imaginary solutions.

No real solutions and two imaginary solutions.

One real solution and no imaginary solutions.

Two real solutions and no imaginary solutions.

Explanation:

Write the equation in standard form:

Evaluate the discriminant , using .

The discriminant is positive, so the equation has two distinct real solutions.

← Previous 1 3

Tired of practice problems?

Try live online GMAT prep today.