### All Precalculus Resources

## Example Questions

### Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

True or false: for a quadratic function of form ax^{2 }+ bx + c = 0, if the discriminant b^{2 }- 4ac = 0, there is exactly one real root.

**Possible Answers:**

False

True

**Correct answer:**

True

This is true. The discriminant b^{2 }- 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b^{2 }- 4ac and evaluate, three cases can happen:

b^{2 }- 4ac > 0

b^{2 }- 4ac = 0

b^{2 }- 4ac < 0

In the first case, having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes , this scenario yields two real results.

In the middle case (the case of our example), . Going back to the quadratic formula , you can see that when everything under the square root is simply 0, then you get only , which is why you have exactly one real root.

For the final case, if b^{2 }- 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

### Example Question #82 : Graphing Functions

True or false: for a quadratic function of form ax^{2 }+ bx + c = 0, if the discriminant b^{2 }- 4ac > 0, there are exactly 2 distinct real roots of the equation.

**Possible Answers:**

True

False

**Correct answer:**

True

This is true. The discriminant b^{2 }- 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b^{2 }- 4ac and evaluate, three cases can happen:

b^{2 }- 4ac > 0

b^{2 }- 4ac = 0

b^{2 }- 4ac < 0

In the first case (the case of our example), having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes , this scenario yields two real results.

In the middle case, . Going back to the quadratic formula , you can see that when everything under the square root is simply 0, then you get only , which is why you have exactly one real root.

For the final case, if b^{2 }- 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

### Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

True or false: for a quadratic function of form ax^{2 }+ bx + c = 0, if the discriminant b^{2 }- 4ac < 0, there are exactly two distinct real roots.

**Possible Answers:**

False

True

**Correct answer:**

False

This is false. The discriminant b^{2 }- 4ac is the part of the quadratic formula that lives inside of a square root function. As you plug in the constants a, b, and c into b^{2 }- 4ac and evaluate, three cases can happen:

b^{2 }- 4ac > 0

b^{2 }- 4ac = 0

b^{2 }- 4ac < 0

In the first case, having a positive number under a square root function will yield a result that is a positive number answer. However, because the quadratic function includes , this scenario yields two real results.

In the middle case, . Going back to the quadratic formula , you can see that when everything under the square root is simply 0, then you get only , which is why you have exactly one real root.

For the final case (the case of our example), if b^{2 }- 4ac < 0, that means you have a negative number under a square root. This means that you will not have any real roots of the equation; however, you will have exactly two imaginary roots of the equation.

### Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b^{2} - 4ac to find the discriminant of the following equation: 4x^{2} + 19x - 5 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

**Possible Answers:**

Discriminant: 281

Two imaginary roots:

Discriminant: 281

Two imaginary roots:

Discriminant: 441

Two real roots: or

Discriminant: 441

Two real roots: or

Discriminant: 0

One real root:

**Correct answer:**

Discriminant: 441

Two real roots: or

In the above equation, a = 4, b = 19, and c = -5. Therefore:

b^{2} - 4ac = (19)^{2} - 4(4)(-5) = 361 + 80 = 441.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two real roots.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

which simplifies to

which gives us two answers:

or

These values of x are the two distinct real roots of the given equation.

### Example Question #91 : Graphing Functions

Use the formula b^{2} - 4ac to find the discriminant of the following equation: 4x^{2} + 12x + 10 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

**Possible Answers:**

Discriminant: 16

Types of Roots: Two distinct real roots

Exact Roots: -1, -2

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots:

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots:

Discriminant: 304

Types of Roots: Two distinct real roots

Exact Roots:

Discriminant: 304

Types of Roots: Two distinct real roots

Exact Roots:

**Correct answer:**

Discriminant: -16

Types of Roots: No real roots; 2 distinct imaginary roots

Exact Roots:

In the above equation, a = 4, b = 12, and c = 10. Therefore:

b^{2} - 4ac = (12)^{2} - 4(4)(10) = 144 - 160 = -16.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two distinct imaginary roots.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

In other words, our two distinct imaginary roots are and

### Example Question #92 : Graphing Functions

Use the formula b^{2} - 4ac to find the discriminant of the following equation: -3x^{2} + 6x - 3 = 0.

Then state how many roots it has, and whether they are real or imaginary. Finally, use the quadratic function to find the exact roots of the equation.

**Possible Answers:**

Discriminant: 72

Two distinct real roots:

Discriminant: 72

Two distinct real roots:

Discriminant: 0

One real root: x = -1

Discriminant: -72

Two distinct imaginary roots:

Discriminant: 0

One real root: x = 1

**Correct answer:**

Discriminant: 0

One real root: x = 1

In the above equation, a = -3, b = 6, and c = -3. Therefore:

b^{2} - 4ac = (6)^{2} - 4(-3)(-3) = 36 - 36 = 0.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, there is exactly one real root.

Finally, we use the quadratic function to find these exact root. The quadratic formula is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

which simplifies to

which gives us one answer: x = 1

This value of x is the one distinct real root of the given equation.

### Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b^{2} - 4ac to find the discriminant of the following equation: x^{2} + 5x + 4 = 0.

**Possible Answers:**

Discriminant: 9

Two real roots: x = -1 or x = -4

Discriminant: 0

One real root:

Discriminant: 41

Two imaginary roots:

Discriminant: 9

Two real roots: x = 1 or x = 4

Discriminant: 41

Two imaginary roots:

**Correct answer:**

Discriminant: 9

Two real roots: x = -1 or x = -4

In the above equation, a = 1, b = 5, and c = 4. Therefore:

b^{2} - 4ac = (5)^{2} - 4(1)(4) = 25 - 16 = 9.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two real roots.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

which simplifies to

which gives us two answers:

x = -1 or x = -4

These values of x are the two distinct real roots of the given equation.

### Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b^{2} - 4ac to find the discriminant of the following equation: -x^{2} + 3x - 3 = 0.

**Possible Answers:**

Discriminant: 0

One real root:

Discriminant: -8

Two imaginary roots:

Discriminant: -21

Two imaginary roots:

Discriminant: -21

Two imaginary roots:

Discriminant: -8

Two imaginary roots: .

**Correct answer:**

Discriminant: -8

Two imaginary roots: .

In the above equation, a = -1, b = 3, and c = -3. Therefore:

b^{2} - 4ac = (3)^{2} - 4(-1)(-3) = 9 - 12 = -3.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two distinct imaginary roots.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

Because , this simplifies to . In other words, our two distinct imaginary roots are and

### Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b^{2} - 4ac to find the discriminant of the following equation: x^{2} + 2x + 10 = 0.

**Possible Answers:**

Discriminant: 36

Two real roots: x = -5 or x = 7

Discriminant: -36

Two imaginary roots:

Discriminant: 0

One real root: x = -1

Discriminant: 36

Two real roots: x = 5 or x = -7

Discriminant: -36

Two imaginary roots:

**Correct answer:**

Discriminant: -36

Two imaginary roots:

In the above equation, a = 1, b = 2, and c = 10. Therefore:

b^{2} - 4ac = (2)^{2} - 4(1)(10) = 4 - 40 = -36.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, we have two distinct imaginary roots.

Finally, we use the quadratic function to find these exact roots. The quadratic function is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

Because , this simplifies to . We can further simplify this to . In other words, our two distinct imaginary roots are and .

### Example Question #1 : Find Roots Of Quadratic Equation Using Discriminant

Use the formula b^{2} - 4ac to find the discriminant of the following equation: x^{2} + 8x + 16 = 0.

**Possible Answers:**

Discriminant: 0

One real root: x = 4

Discriminant: 72

Two distinct real roots:

Discriminant: 0

One real root: x = 0

Discriminant: 128

Two distinct real roots:

Discriminant: 0

One real root: x = -4

**Correct answer:**

Discriminant: 0

One real root: x = -4

In the above equation, a = 1, b = 8, and c = 16. Therefore:

b^{2} - 4ac = (8)^{2} - 4(1)(16) = 64 - 64 = 0.

When the discriminant is greater than 0, there are two distinct real roots. When the discriminant is equal to 0, there is exactly one real root. When the discriminant is less than zero, there are no real roots, but there are exactly two distinct imaginary roots. In this case, there is exactly one real root.

Finally, we use the quadratic function to find these exact root. The quadratic formula is:

Plugging in our values of a, b, and c, we get:

This simplifies to:

which simplifies to

This value of x is the one distinct real root of the given equation.