Algebra: 9.1.4 Choosing a Method
Bell Work 2/3
a.
Solutions
Name ____________________________
Block ____ Date _________
Solve each quadratic equation using the quadratic formula:
𝑥 2 + 3𝑥 − 5 = 0
b.
−3 ± �32 − 4(1)(−5)
𝑥=
2(1)
−3 ± √9 + 20
2
𝑥=
𝒙=
−𝟑 ± √𝟐𝟐
𝟐
𝑥=
3𝑥 2 + 2𝑥 − 4 = 0
𝑥=
−𝑏 ± √𝑏2 − 4𝑎𝑎
2𝑎
−2 ± �22 − 4(3)(−4)
2(3)
𝑥=
−2 ± √4 + 48
6
−2 ± √52
6
𝑥=
𝑥=
−2 ± √4 ∙ 13
6
𝑥=
𝒙=
−2 ± 2√13
6
−𝟏 ± √𝟏𝟏
𝟑
1. Why is completing the square a reasonable strategy for solving these two equations?
There is only one x2 and an even number of x’s in each polynomial
Solve each quadratic Equation by Completing the Square:
a. x2 + 12x + 27 = 0
𝑥 2 − 12𝑥 = −27
b. x2 + 5 − 2x = 0
𝑥 2 − 2𝑥 = −5
𝑥 2 − 12𝑥 + 36 = −27 + 36
𝑥 2 − 2𝑥 + 1 = −5 + 1
𝑥 − 6 = ±3
𝑥 − 1 = ±√−4
(𝑥 − 6)2 = 9
𝑥 = 6±3
(𝑥 − 1)2 = −4
No Real Roots!
𝑥 = 𝟗 and 𝒙 = 𝟑
2. Why is using factors (and zero product property) a reasonable strategy for solving these equations?
Solve each equation by Factoring:
a. (3x + 4)(2x − 1) = 0
3𝑥 + 4 = 0
3𝑥 = −4
−𝟒
𝑥=
𝟑
2𝑥 − 1 = 0
2𝑥 = 1
𝟏
𝑥=
𝟐
One is already factored and the other contains no decimals or fractions
b. 20x2 − 30x = 2x + 45
20𝑥 2 − 32𝑥 − 45 = 0
2𝑥 − 5 = 0
2𝑥 = 5
𝑥=
𝟓
𝟑
10𝑥 + 9 = 0
10𝑥 = −9
𝑥=
−𝟗
𝟏𝟏
−5 −50𝑥 −45
2𝑥 20𝑥 2
10𝑥
18𝑥
+9
3. Why is using the Quadratic Formula a reasonable strategy for solving these equations?
The Quadratic Formula can be used with any quadratic equation
Solve each equation by using the Quadratic Formula:
a. 0.5x2 + 9x − 2.8 = 6
0.5𝑥 2 + 9𝑥 − 8.8 = 0
𝑥=
−9 ± �(9)2 − 4(.5)(−2.8)
2(.5)
𝑥=
9 ± √81 + 5.6
1
𝒙 = 𝟗 ± √𝟖𝟖. 𝟔
𝑥 ≈ 18.3
𝑥 ≈ −0.3
b. x2 + 5 − 2x = 0
𝑥 2 − 2𝑥 + 5 = 0
𝑥=
2 ± �(−2)2 − 4(1)(5)
2(1)
𝑥=
2 ± √4 − 20
2
𝑥=
2 ± √−16
2
No Real Roots!
4.
A Rational number can be written as a ratio. All integers can be written as a fraction with a denominator of 1. Use
the calculator to convert each of the following into an integer or fraction (if possible) using: mee.
• Write Rational next to those you can convert to a fraction or integer
• Write Irrational next to those numbers you cannot convert to a fraction or integer.
π
3.14
Irrational
157
50
Rational
√5
Irrational
0.045
9
200
3
√ 0.0036
Rational
√0.124
3
50
Irrational
Rational
9-36. Consider the equation: (x − 5)(x + 2) = −6
a. What is wrong with this equation? It is not equal to zero
b. What must you do before you can solve this equation? Multiply to simplify and add six to both sides
c. Solve the equation using any method. (Show sufficient work).
−5 −5𝑥
𝑥
𝑥2
𝑥
−10
𝑥 2 − 3𝑥 − 10 = −6
+2
𝑥 = −𝟏 and 𝒙 = 𝟒
2𝑥
𝑥 2 − 3𝑥 − 4 = 0
9-37. MOE’S YO: Moe is playing with a yo-yo. He throws the yo-yo down and then
pulls it back up. The motion of the yo-yo is represented by y = 2x2 −4.8x,
where x represents the number of seconds since the yo-yo left Moe’s hand,
and y represents the vertical height in feet of the yo-yo with respect to Moe’s hand.
Note that when the yo-yo is in Mo’s hand, y = 0, and when the yo-yo is below his
hand, y is negative.
a.
How long is Moe’s yo-yo in the air before it comes back to Moe’s hand? Write
and solve a quadratic equation to find the times that the yo-yo is in Moe’s hand.
b.
How long does it take for the yo-yo to turn around, that is, to start its return to his
hand? Use what you know about parabolas to help you.
c.
How long is the yo-yo’s string? That is, what is y when the yo-yo changes
direction?
d.
Sketch a graph representing the motion of Moe’s yo-yo on the axes to the
right. On the sketch, label axes and label all important points for this situation
2.4
0
x
(1.2, -2.88)
y
5. Simplify the following radical expressions (show intermediate steps as necessary)
a. √72 = √36 ∙ 2 = 𝟔√𝟐
b. 2√700 = 2√100 ∙ 7 = 2 ∙ 10√7 = 𝟐𝟐√𝟕
c. 3√98 = 3√49 ∙ 2 = 3 ∙ 7√2 = 𝟐𝟐√𝟐
d. √121 = 11
e. √24 = √4 ∙ 6 = 𝟐√𝟔
6. Write equations in graphing, standard and factored form for the quadratic function that is graphed.
10
y
8
2
Graphing: 𝑦 = (𝑥 − 1) − 4
6
4
Standard: 𝑦 = 𝑥 2 − 2𝑥 − 3
2
Factored: 𝑦 = (𝑥 + 1)(𝑥 − 3)
x
−10
−8
−6
−4
−2
2
4
6
8
10
−2
−4
−6
−8
−10
9-38. Write and solve a system of equations for the situation described below. Define your variables
and write your solution as a sentence.
Daria has all nickels and quarters. The number of nickels is 3 more than twice the
number of quarters. If she has $1.90 in all, how many nickels does Daria have?
N = the number of nickels
Q = the number of quarters
𝑁 = 2𝑄 + 3
0.05𝑁 + 0.25𝑄 = 1.90
0.05(2𝑄 + 3) + 0.25𝑄 = 1.90
𝑁 = 2𝑄 + 3
0.1𝑄 + 0.15 + 0.25𝑄 = 1.90
𝑁 = 2(5) + 3
0.35𝑄 = 1.75
𝑁 = 13
0.35𝑄 + 0.15 = 1.90
𝑄=5
There are 13 Nickels in Daria’s coin collection.
𝑁 = 10 + 3
9-39. Solve the following quadratic equations using any method (show sufficient work).
a.
10000x2 − 64 = 0
b.
9x2 − 8 = −34x
c.
2x2 − 4x + 7 = 0
d.
3.2x + 0.2x2 − 5 = 0
9-40. Write an equation to represent the number of tiles, y, in Figure x for the tile pattern.