Algebra II
Name:
Solving Quadratics by Factoring
NOTES
Vocabulary Check
graph of a quadratic function is called a
other terms for x-intercepts are
number of solutions to a quadratic equation:
when the vertex is the highest point on the graph of a quadratic, it is called a
when the vertex is the lowest point on the graph of a quadratic, it is called a
The Zero Product Property
1.List possible solutions for a and b in a ⋅ b = 24 2.List possible solutions for a and b in a ⋅ b = 0
3. Solve the equations using The Zero Product Property. NO CALCULATOR
a. (3x + 6)(2x + 5) = 0
b. (x – 7)2 = 0
c. (2x + 1)(3x – 2)(x – 1) = 0
Solving Quadratics by Factoring
Solve each equation. NO CALCULATOR
1) x2 + 7x + 15 = 5
2) x2 = –3x + 18
3) 12x2 – 5x = 0
4) 49x2 = –100
5) 4x3 – 22x2 – 42x = 0
Connection to Graphing Quadratics
1) What step was done to change y = x 2 − 4 x − 5 into 0 = x 2 − 4 x − 5 ?
2) Put y = x 2 − 4 x − 5 into factored form.
What does this tell us about the graph?
3) Solve 0 = x 2 − 4 x − 5 by factoring.
Conclusion:
**Remember: A table can also be helpful in finding real solutions to equations.
3) Check your answer to 3.c. (2x + 1)(3x – 2)(x – 1) = 0 (from The Zero Product Property
section) using a graphing calculator.
4) Sketch a quick graph of 3) 12x2 – 5x = 0 (from the Solving Quadratics by Factoring Section).
Algebra II
Name:
Solving Quadratics by Factoring
HOMEWORK
NO CALCULATOR (#1 – 13)
Solve each equation.
1) (4x – 8)(7x + 21) = 0
2) 7x2 – 35x = 0
3) x2 + 5x – 35 = 3x
4) 20x2 = –15x
5) x2 + 16 = 0
6) 5(3x + 9)(5x – 15) = 0
7) 8x2 + 21 = –59x
8) (2x + 3)2 – 36 = 0
9) 6x2 + 33x + 42 = 0
10) Sketch a graph of y = −2 ( x − 4 ) − 3 .
2
By looking at this graph, what can you say about the solutions to −2 ( x − 4 ) − 3 = 0 ?
2
Sketch a possible graph of the quadratic function that satisfies the given conditions.
11) The vertex of the parabola is at (4, –3) and one of the x-intercepts is 6.
12) The parabola has x-intercepts of 2 and –2.
13) One root of a quadratic function is between –4 and –3, and the other root is between 1 and 2.
The maximum point of the function is at (–1, 6).
CALCULATOR ALLOWED (#14-20)
14) Check your answers to #1-9 on a graphing calculator. Correct any problems that you missed.
That doesn’t mean just change your final answer! Find the mistake in your work and correct
the problem so that the answers match those that you found on your calculator.
15
15)
16)
16
An art gallery has walls that are sculpted with arches that can be represented by the quadratic
function f (x) = –x2 – 4x + 12, where x is in feet. The wall space under each arch is to be painted
a different color from the arch itself.
17) Graph the quadratic function and determine its x-intercepts.
18) What is the length of the segment along the floor of each arch?
19) What is the maximum height of the arch?
2
20) The formula A = bh can be used to estimate the area under a parabola. In this formula, A
3
represent area, b represents the length of the base, and h represents the height. Calculate that area
that needs to be painted for one parabola.
BONUS! How much would the paint for the walls under 12 arches cost if the paint is $27 per
gallon, the painter applies 2 coats, and the manufacturer states that each gallon will cover 200
square feet? (Hint: Remember that you cannot buy part of a gallon.) SHOW ALL WORK!