Solutions
Algebra: 8.2.2 Solving Quadratic Equations Name ______________________________
Block _____ Date ______
Bell Work 1/15 Factor each quadratic equation completely. Show sufficient work.
a. y = 2x2 + 8x + 8
2
𝑦 = 2(𝑥 + 4𝑥 + 4)
𝑦 = 𝟐( 𝒙 + 𝟐) 𝟐
2 2𝑥
𝑥 𝑥2
𝑥
4
b. y = 25x2 − 64
𝑦 = (𝟓𝒙 + 𝟖)(𝟓𝒙 − 𝟖)
2𝑥
2
8-64. Graph y = x − 2x – 15 without using the calculator. (Adjust scale as needed)
2
3 3𝑥 −15
𝑥 𝑥
2
𝑥
a. Factored form:
−5𝑥
−5 b. Find the roots:
Explain how to find.
c. Find the y-intercept.
Explain how to find.
𝑦 = (𝑥 + 3)(𝑥 − 5)
𝑥+3=0
𝑥−5=0
𝑥 = −3
𝑥=5
Set Factors equal to 0 and solve
-3
𝑦 = −15
It is the last number in
the standard equation
d. Find the x-coordinate of the vertex.
Explain how to find.
−3 + 5
=1
2
Average the roots
5
-15
(1,-16)
𝑦 = (1 + 3)(1 − 5)
𝑦 = (4)(−4)
𝑦 = −16
Plug x into an equation and solve
2
8-65. Graph y = 2x + 5x – 12 without using the calculator.
e. Find the y-coordinate of the vertex.
Explain how to find.
4 8𝑥 −12
𝑥 2𝑥 2 −3𝑥 a. Factored form:
2𝑥
−3 b. Find the roots:
𝒚 = (𝒙 + 𝟒)(𝟐𝟐 − 𝟑)
𝑥+4 =0
𝒙 = −𝟒
2𝑥 − 3 = 0
2𝑥 = 3
𝒙 = 𝟏. 𝟓
c. Find the y-intercept.
𝒚 = −𝟏𝟏
d. Find the x-coordinate of the vertex. −4 + 1.5
𝑥=
= −1.25
2
e. Find the y-coordinate of the vertex.
𝑦 = (−1.25 + 4)(2(−1.25) − 3)
𝑦 = (2.25)(−5.5) = −15.25
-4
1.5
-12
(-1.25, -15.25)
8-68. Graph y = x2 – 2x – 8 without using the calculator.
2 2𝑥
−8
𝑥 𝑥 2 −4𝑥 a. Factored form:
𝑥
−4 b. Find the roots:
𝒚 = (𝒙 + 𝟐)(𝒙 − 𝟒)
𝒙 = −𝟐
c. Find the y-intercept. 𝒚 = −𝟖
𝒙=𝟒
-2
d. Find the x-coordinate of the vertex. −𝟐 + 𝟒
𝒙=
=𝟏
𝟐
e. Find the y-coordinate of the vertex.
𝒚 = 𝟏𝟐 − 𝟐(𝟏) − 𝟖 = −𝟗
4
-8
(1,-9)
8-69. Graph y = x2 + x – 6 without using the calculator.
3 3𝑥
−6
𝑥 𝑥 2 −2𝑥 a.
𝑥
Factored form:
−2 b.
Find the roots:
𝒚 = (𝒙 + 𝟑)(𝒙 − 𝟐)
𝒙 = −𝟑
𝒙=𝟐
𝒚 = −𝟔
c.
Find the y-intercept.
d.
Find the x-coordinate of the vertex.
−𝟑 + 𝟐
𝒙=
= −𝟎. 𝟓
Find the y-coordinate of the vertex. 𝟐
e.
𝒚 = (−𝟎. 𝟓 + 𝟑)(−𝟎. 𝟓 − 𝟐)
𝒚 = (𝟐. 𝟓)(−𝟐. 𝟓)
𝒚 = −𝟔. 𝟐𝟐
8-70. Compare the two equations below.
(x + 2)(x − 1) = 0 and (x + 2) + (x − 1) = 0
a. How are the equations different?
The first equation is multiplying
binomials (quadratic). The second
equation is adding binomials (linear)
b. Solve both equations.
(𝑥 + 2)(𝑥 − 2) = 0
𝑥+2 =0
𝑥−1 =0
𝒙 = −𝟐
𝒙=𝟏
2 Solutions
𝑥+2+𝑥−1= 0
2𝑥 + 1 = 0
2𝑥 = −1
−𝟏
𝒙=
𝟐
1 Solution
The following are answers only. Show SUFFICIENT WORK!.
8-71. For each equation below, solve for x.
a.
(x − 2)(x + 8) = 0
b.
(3x − 9)(x − 1) = 0
c.
(x + 10)(2x − 5) = 0 𝒙 = −𝟏𝟏
(x − 7)2 = 0
d.
𝒙=𝟐
𝒙 = −𝟖
𝒙=𝟑
𝒙=𝟕
𝒙=𝟏
𝒙=
𝟓
𝟐
8-72. Solve the system of equations.
5x − 2y = 4
x=0
(𝟎, −𝟐)
8-73. The (x-intercepts) of the graph of y = 2x2 − 16x + 30 are (3, 0) and (5, 0).
a.
b.
What is the x-coordinate of the vertex? How do you know?
4, it is halfway between 3 and 5.
Use your answer to part (a) above to find the y-coordinate of the vertex. Then
write the vertex as a point (x, y).
𝑦 = 2(4)2 − 16(4) + 30 = −2
8-74. Factor each quadratic below completely.
a.
2x2 − 2x − 4
b.
4x2 − 24x + 36 = 𝟒(𝒙 − 𝟑)𝟐
= 𝟐(𝒙 − 𝟐)(𝒙 + 𝟏)
EOC Practice:
−𝟒
𝟒
𝟏𝟏
𝟏𝟏
(𝟒, −𝟐)