Introduction to Quadratic Equations

Show your work for each problem.

1.
Which of the following is a quadratic equation?
1. a)
2. b)
3. c)
4. d)
Answer:
is quadratic because the highest power of is 2. The first is linear (degree 1), the third is cubic (degree 3), and the fourth is not a polynomial.
2.
In the equation , what is the value of ?
Answer: 2
Comparing with , the coefficient of is 2, so .
3.
For the equation , what is the value of ?
1. a)
2. b)
3. c)
4. d)
Answer:
Write as . The coefficient of is , so . A common mistake is forgetting the negative sign.
4.
In , what is the value of ?
Answer: -6
In , the constant term is , so . Remember to include the negative sign!
5.
If in a quadratic equation, which direction does the parabola open?
1. a)Left
2. b)Right
3. c)Upward (∪)
4. d)Downward (∩)
Answer: Upward (∪)
Since (positive), the parabola opens upward. If were negative, it would open downward.
6.
Rewrite in standard form and identify , , and .
Answer: a=1, b=-3, c=-10
1. Subtract from both sides. What do you get? 0 = x^2 - 3x - 10
2. Rewrite with the quadratic term first: . What is ? 1
3. What is ? -3
4. What is ? -10
7.
Find the x-coordinate of the vertex for .
Answer: 3
With and : . The vertex is at .
8.
Find the vertex of .
Answer: (-2, -9)
1. Identify and . a=1, b=4
2. Calculate . -2
3. Find by substituting into the equation: . -9
4. Write the vertex as an ordered pair. (-2, -9)
9.
For , the vertex represents:
1. a)A maximum point
2. b)The y-intercept
3. c)A minimum point
4. d)The x-intercept
Answer: A maximum point
Since , the parabola opens downward (∩ shape), so the vertex is the highest point - a maximum.
10.
If the vertex of a parabola is at and , is 5 the maximum or minimum value? (Write 'maximum' or 'minimum')
Answer: maximum
When , the parabola opens downward, making the vertex the highest point. Therefore, is the maximum value.
11.
For , find the axis of symmetry and vertex.
Answer: x=2, vertex (2, -2)
1. What are and ? a=2, b=-8
2. Find the axis of symmetry: 2
3. Calculate -2
4. Write the vertex. (2, -2)
12.
A ball is thrown upward. Its height is . When does it reach maximum height?
1. a) second
2. b) seconds
3. c) seconds
4. d) seconds
Answer: seconds
Maximum height occurs at the vertex. seconds.
13.
A company's profit is euros, where is items sold. Find the number of items to maximize profit.
Answer: 30
1. Is this a maximum or minimum problem? (Look at the sign of ) maximum
2. What is ? 120
3. Calculate 30
14.
For , what is the maximum value of the function?
Answer: 3
Vertex x: . Maximum value: .
15.
Rewrite in standard form, identify , , , and find the vertex.
Answer: (-1, -16)
1. Subtract 15 from both sides. Write the standard form. x^2 + 2x - 15 = 0
2. What are , , and ? a=1, b=2, c=-15
3. Find the x-coordinate of the vertex: -1
4. Find the y-coordinate: -16
5. Write the vertex as an ordered pair. (-1, -16)