Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Math 1010
Section 8.1: Solving Quadratic Equations: Factoring and Special Forms
Ex.1
Solve each quadratic equation by factoring.
(1) x2 + 5x = 24
(2) 3x2 + 11x − 4 = 0
(3) 9x2 + 12 = 3 + 12x + 5x2
Square root property
The equation u2 = d, where d > 0, has exactly two solutions:
extracting square roots.
1
√
√
d and − d. The solution process is called
Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Math 1010
Ex.2
Solve each quadratic equation.
(1) 3x2 = 15
(2) (x − 2)2 = 10
(3) (3x − 6)2 − 8 = 0
Complex square root property
The equation u2 = d, where d < 0, has exactly two solutions:
Ex.3
Solve each equation.
(1) x2 + 8 = 0
(2) (x − 4)2 = −3
(3) 2(3x − 5)2 + 32 = 0
2
p
p
|d|i and − |d|i.
Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Equation of quadratic form
An equation is of quadratic form if it has the form
au2 + bu + c = 0
where u is an algebraic expression.
Ex.4
Solve x4 − 13x2 + 36 = 0.
Ex.5
√
Solve x − 5 x + 6 = 0.
3
Math 1010
Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Math 1010
Section 8.2: Completing the Square
Completing the square
To complete the square for the expression x2 + bx, add (b/2)2 , which is the square of half the coefficients of
x. Therefore,
b 2 b 2
x2 + bx +
= x+
2
2
Ex.1
Solve x2 + 12x = 0 by completing the square.
Ex.2
Solve x2 − 6x + 7 = 0 by completing the square.
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Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Ex.3
Solve 2x2 − x − 2 = 0 by completing the square.
Ex.4
Solve 3x2 − 6x + 1 = 0 by completing the square.
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Math 1010
Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Math 1010
Ex.5
Solve x2 − 4x + 8 = 0 by completing the square.
Ex.6
An open box with a rectangular base of 2x inches by 6x − 2 inches has a height of 9 inches. The volume
of the box is 1584. Find the dimensions of the box.
6
Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Section 8.3: The Quadratic Formula
The quadratic formula
The solutions of ax2 + bx + c = 0, a 6= 0, are given by the Quadratic Formula
√
−b ± b2 − 4ac
x=
2a
Ex.1
Solve x2 + 6x = 16.
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Math 1010
Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Ex.2
Solve −x2 − 4x + 8 = 0.
Ex.3
Solve 18x2 − 24x + 8 = 0.
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Math 1010
Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Math 1010
Ex.4
Solve 2x2 − 4x + 5 = 0.
Section 8.4: Graphs of Quadratic Functions
Graphs of quadratic functions
The graph of f (x) = ax2 + bx + c, a 6= 0, is a parabola. The completed square form
f (x) = a(x − h)2 + k
is the standard form of the function. The vertex of the parabola occurs at the point (h, k), and the vertical
line passing through the vertex is the axis of the parabola.
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Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Ex.1
Find the vertex of the parabola given by f (x) = x2 − 6x + 5.
Ex.2
Find the vertex of the parabola given by f (x) = 3x2 − 9x.
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Math 1010
Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Math 1010
Sketching the graph of a parabola
In order to sketch the graph of a parabola you need to follow these steps:
(1) Determine the vertex of the parabola by completing the square or by using the formula x = −b/(2a).
(2) Plot the vertex, axis, x and y intercepts, and a few additional points on the parabola.
(3) Use the fact that the parabola opens upward is a > 0 and opens downward if a < 0 to complete the
sketch.
Ex.3
Sketch the graph of the parabola given by f (x) = x2 + 6x + 8.
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Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Math 1010
Ex.4
Write the equation of the parabola with vertex (−2, 1) and y-intercept (0, −3) (its graph opens downward).
Ex.5
Write the equation of the parabola with vertex (3, −4) and that passes through the point (5, −2) (its graph
opens upward).
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Chapter 8: Quadratic Eq, Functions, and Ineq.Lecture notes
Math 1010
Section 8.5: Applications of Quadratic Functions
Ex.1
Find the two consecutive positive integers such that their product is 156.
Ex.2
An L-shaped sidewalk from the athletic center to the library on a college campus is 200 meters long. By
cutting diagonally across the grass, students shorten the walking distance to 150 meters. What are the
two lengths of the sidewalk?
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