Quadratic

Question 1

What is the standard form of a quadratic equation?

Answer 1

ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0.

Question 2

True or False: A quadratic equation can have at most two real solutions.

Answer 2

True.

Question 3

What method can be used to solve a quadratic equation?

Answer 3

Factoring, completing the square, or using the quadratic formula.

Question 4

Fill in the blank: The quadratic formula is x = __________.

Answer 4

(-b ± √(b² - 4ac)) / (2a)

Question 5

What does the discriminant (b² - 4ac) indicate?

Answer 5

The nature of the roots of the quadratic equation.

Question 6

If the discriminant is positive, how many real solutions does the quadratic equation have?

Answer 6

Two distinct real solutions.

Question 7

If the discriminant is zero, how many real solutions does the quadratic equation have?

Answer 7

One real solution (a repeated root).

Question 8

If the discriminant is negative, how many real solutions does the quadratic equation have?

Answer 8

No real solutions (two complex solutions).

Question 9

What is the vertex form of a quadratic equation?

Answer 9

y = a(x - h)² + k, where (h, k) is the vertex.

Question 10

True or False: The vertex of a quadratic function can be found using the formula h = -b/(2a).

Answer 10

True.

Question 11

What is the axis of symmetry for a quadratic equation in standard form?

Answer 11

x = -b/(2a).

Question 12

How can you determine if a quadratic is factorable?

Answer 12

By checking if the discriminant is a perfect square.

Question 13

What are the roots of the equation x² - 5x + 6 = 0?

Answer 13

x = 2 and x = 3.

Question 14

What is the value of a in the equation 3x² + 4x + 1 = 0?

Answer 14

3.

Question 15

What is the value of c in the equation x² - 4x + c = 0 if it has one solution?

Answer 15

c = 4.

Question 16

True or False: Completing the square can always be used to solve any quadratic equation.

Answer 16

True.

Question 17

What is the first step in completing the square for the equation x² + 6x + 5 = 0?

Answer 17

Rewrite it as x² + 6x = -5.

Question 18

What is the solution to the equation 2x² - 8 = 0?

Answer 18

x = ±2.

Question 19

Which method is generally preferred for solving quadratic equations when coefficients are large?

Answer 19

Using the quadratic formula.

Question 20

What are the roots of the equation 4x² + 12x + 9 = 0?

Answer 20

x = -3 (a repeated root).

Question 21

Fill in the blank: The graph of a quadratic equation is a __________.

Answer 21

parabola.

Question 22

What does the coefficient ‘a’ determine about the parabola?

Answer 22

The direction of opening (upwards if a > 0, downwards if a < 0).

Question 23

What is the maximum value of the quadratic function y = -2x² + 8x - 5?

Answer 23

The maximum value is 7.

Question 24

True or False: The roots of a quadratic equation can be complex numbers.

Answer 24

True.

Question 25

How can you find the y-intercept of a quadratic equation?

Answer 25

By evaluating the equation at x = 0.

Question 26

What is a quadratic equation?

Answer 26

A quadratic equation is a math problem in the form of ax² + bx + c = 0.

Question 27

What does 'a' represent in a quadratic equation?

Answer 27

'a' is the number in front of x².

Question 28

What does 'b' represent in a quadratic equation?

Answer 28

'b' is the number in front of x.

Question 29

What does 'c' represent in a quadratic equation?

Answer 29

'c' is the constant number without x.

Question 30

True or False: Quadratic equations can have two solutions.

Answer 30

True.

Question 31

What is the first step in solving a quadratic equation?

Answer 31

Make sure the equation is in the form ax² + bx + c = 0.

Question 32

What is factoring?

Answer 32

Factoring is breaking a number or expression into parts that multiply together.

Question 33

Fill in the blank: The quadratic formula is x = ___.

Answer 33

(-b ± √(b² - 4ac)) / (2a)

Question 34

What do you find inside the square root in the quadratic formula?

Answer 34

You find b² - 4ac, called the discriminant.

Question 35

What does the discriminant tell you?

Answer 35

It tells you how many solutions the quadratic equation has.

Question 36

True or False: A negative discriminant means there are no real solutions.

Answer 36

True.

Question 37

What are the two types of solutions for a quadratic equation?

Answer 37

Real solutions and complex solutions.

Question 38

What is completing the square?

Answer 38

Completing the square is a method to solve a quadratic by rewriting it in a perfect square form.

Question 39

What is the vertex of a parabola?

Answer 39

The vertex is the highest or lowest point of the parabola.

Question 40

What shape does a quadratic equation graph make?

Answer 40

A parabola.

Question 41

True or False: All quadratic equations can be solved by factoring.

Answer 41

False.

Question 42

What do you do if you cannot factor a quadratic equation easily?

Answer 42

Use the quadratic formula.

Question 43

What is the axis of symmetry in a quadratic equation?

Answer 43

The line that divides the parabola into two mirror images.

Question 44

How do you find the axis of symmetry?

Answer 44

Use the formula x = -b / (2a).

Question 45

What does the '±' symbol in the quadratic formula mean?

Answer 45

It means you will have two possible values for x.

Question 46

Fill in the blank: The solutions of a quadratic equation are also called ___.

Answer 46

roots.

Question 47

What is the maximum number of real solutions a quadratic can have?

Answer 47

Two.

Question 48

What happens if the discriminant is zero?

Answer 48

There is exactly one real solution.

Question 49

True or False: You can always graph a quadratic equation.

Answer 49

True.

Question 50

What do you call the highest point of a downward-opening parabola?

Answer 50

Maximum.

Question 51

What do you call the lowest point of an upward-opening parabola?

Answer 51

Minimum.

Question 52

What is the first step to use when trying to solve by completing the square?

Answer 52

Move the constant term to the other side of the equation.