Topic 6 Solving Problems Using Quadratic Equations

Question 1

What is the standard form of a quadratic equation?

Answer 1

ax^2 + bx + c = 0

Question 2

True or False: The highest power of the variable in a quadratic equation is 2.

Answer 2

True

Question 3

What are the coefficients in the quadratic equation ax^2 + bx + c?

Answer 3

a, b, and c

Question 4

Fill in the blank: In a quadratic equation, the value of ‘a’ must be ______.

Answer 4

non-zero

Question 5

What method can be used to solve quadratic equations?

Answer 5

Factoring, completing the square, or using the quadratic formula

Question 6

What is the quadratic formula?

Answer 6

x = (-b ± √(b² - 4ac)) / (2a)

Question 7

True or False: The discriminant is given by the expression b² - 4ac.

Answer 7

True

Question 8

What does a positive discriminant indicate about the roots of a quadratic equation?

Answer 8

Two distinct real roots

Question 9

What does a discriminant of zero indicate?

Answer 9

One real root (a repeated root)

Question 10

What does a negative discriminant indicate?

Answer 10

No real roots (two complex roots)

Question 11

Fill in the blank: The vertex form of a quadratic equation is ______.

Answer 11

y = a(x - h)² + k

Question 12

In the vertex form, what do ‘h’ and ‘k’ represent?

Answer 12

The coordinates of the vertex of the parabola

Question 13

How do you find the vertex from the standard form of a quadratic equation?

Answer 13

Use the formula h = -b/(2a) and then substitute h into the equation to find k.

Question 14

What is the axis of symmetry for a quadratic equation?

Answer 14

x = -b/(2a)

Question 15

True or False: The graph of a quadratic equation is a straight line.

Answer 15

False

Question 16

What shape does the graph of a quadratic equation form?

Answer 16

A parabola

Question 17

What is the effect of a > 0 on the parabola’s opening?

Answer 17

It opens upwards.

Question 18

What is the effect of a < 0 on the parabola’s opening?

Answer 18

It opens downwards.

Question 19

If the quadratic equation is written in factored form, what does it look like?

Answer 19

y = a(x - r1)(x - r2)

Question 20

What do r1 and r2 represent in the factored form of a quadratic equation?

Answer 20

The roots of the equation

Question 21

Fill in the blank: The sum of the roots of the quadratic equation ax^2 + bx + c = 0 is ______.

Answer 21

-b/a

Question 22

Fill in the blank: The product of the roots of the quadratic equation ax^2 + bx + c = 0 is ______.

Answer 22

c/a

Question 23

What is the relationship between the coefficients and the roots of a quadratic equation?

Answer 23

The sum and product of the roots are related to the coefficients.

Question 24

True or False: Quadratic equations can have at most two real solutions.

Answer 24

True

Question 25

Which method is typically easiest for solving simple quadratic equations?

Answer 25

Factoring

Question 26

What type of problems can be solved using quadratic equations?

Answer 26

Projectile motion, area problems, and optimization problems.

Question 27

Fill in the blank: The graph of a quadratic equation can be used to find the maximum or minimum value of the function at the ______.

Answer 27

vertex

Question 28

What is the formula for the vertex's y-coordinate in standard form?

Answer 28

k = c - (b²/(4a))

Question 29

True or False: The roots of a quadratic equation can be found graphically by identifying the x-intercepts.

Answer 29

True

Question 30

What does it mean if the quadratic equation has complex roots?

Answer 30

The graph does not intersect the x-axis.

Question 31

Fill in the blank: To complete the square, you need to transform the equation into the form ______.

Answer 31

(x - p)² = q

Question 32

What is the first step in completing the square for the equation ax^2 + bx + c = 0?

Answer 32

Divide all terms by a.

Question 33

What is the significance of the vertex in a quadratic function?

Answer 33

It represents the maximum or minimum point of the graph.

Question 34

Fill in the blank: The term 'quadratic' comes from the Latin word 'quadratus,' which means ______.

Answer 34

square

Question 35

True or False: All quadratic equations can be solved using the quadratic formula.

Answer 35

True

Question 36

Which type of quadratic equation has no real solutions?

Answer 36

When the discriminant is negative.

Question 37

What is the significance of the coefficient 'a' in the quadratic equation?

Answer 37

It determines the width and direction of the parabola.

Question 38

Fill in the blank: The maximum or minimum value of a quadratic function occurs at the ______.

Answer 38

vertex

Question 39

What is the relationship between the roots and the factorization of a quadratic equation?

Answer 39

The roots can be used to write the equation in factored form.