1.3. QUADRATIC EQUATIONS 29
Problem 1.57 For each of the following quadratic equations, find the discriminant D of the
equation and state the number of roots the equation has.
1. y D x
2
5x C 6
2. y D x
2
4x C 5
3. y D x
2
C 6x C 9
4. y D 2x
2
x 3
5. y D 2x
2
x C 2
6. y D 5 x
2
7. y D x
2
C x C 1
8. y D x
2
C x 1
Problem 1.58 Factor each of the following quadratics. Not all of them factor without a bit of
fiddling.
1. y D x
2
5x C 6
2. y D x
2
C 7x C 12
3. y D 2x
2
C 6x C 4
4. y D 4 C 4x x
2
5. y D .x 1/.x C 1/ 3
6. y D x
2
25
7. y D x
2
3
8. y D x
2
x 1
Problem 1.59 For each of the following quadratics, find all the roots or give a reason there are
none.
1. y D x
2
C 5x C 6
2. y D x
2
C x 6
3. y D x
2
36
4. y D x
2
20x C 91
5. y D x
2
C x C 1
6. y D x
2
C x 1
7. y D x
2
C 36
8. y D 20x
2
C 9x C 1
Problem 1.60 For each of following quadratics, complete the square.
1. y D x
2
2x 2
2. y D x
2
C 4x 3
3. y D 4x
2
C 4x 2
4. y D x
2
x
3
4
5. y D x
2
C 2x C 2
6. y D x
2
C 6x C 10
7. y D x
2
C x C 1
8. y D x
2
C 3x C 1
30 1. REVIEW OF ALGEBRA
Problem 1.61 If the height of a ball at time t in meters is given by
h D 30 C 8t 5t
2
find (without calculus) the time that ball attains its greatest height and the number of meters
above the ground that the ball is at that time.
Problem 1.62 If the height of a ball at time t in meters is given by
h D 20 C 16t 5t
2
find (without calculus) the time that ball attains its greatest height and the number of meters
above the ground that the ball is at that time.
Problem 1.63 If the height of a ball at time t in meters is given by
h D 24 C 4t 5t
2
find (without calculus) the time that ball attains its greatest height and the number of meters
above the ground that the ball is at that time.
Problem1.64 Find a quadratic equation y D ax
2
C bx C c with a, b, and c all whole numbers
that has two roots, neither of which is a whole number divisor of c.
Problem 1.65 Find all values of q for which y D x
2
qx C 2 has two roots.
Problem 1.66 Find all values of q for which y D x
2
3x C q has two roots.
Problem 1.67 Find a quadratic y D ax
2
C bx C c so that the equation has only one root at
x D 2 and so that, when you plug in 3 for x, y D 2.
Problem 1.68 Find two quadratic equations whose graphs have no points in common. Explain
why your solution is correct.
Problem 1.69 Find a quadratic equation that passes through the points (-1,0), (0,1), and (1,0).
Problem 1.70 Find a quadratic equation that passes through the points (0,3), (1,2), and (2,3).
1.3. QUADRATIC EQUATIONS 31
Problem 1.71 Find a quadratic equation that passes through the points (2,-1), (3,4), and (4,9).
Problem 1.72 Suppose we have two points with distinct x coordinates. How many quadratic
equations have graphs that include those two points?
Problem 1.73 Find and graph a quadratic equation that has roots at x D 2; 3.
Problem 1.74 Find a quadratic equation with a root at x D 2 that also passes through the
point (1,1).
Problem1.75 Find a quadratic equation with a root at x D 3 that also passes through the point
(-1,2).
Problem1.76 Find a quadratic equation with a root at x D 4 that also passes through the point
(1,6).
Problem 1.77 Suppose we have three points on the same line. Can a quadratic equation pass
through all three of those points?
Problem 1.78 Consider sets of three points that are not on the same line. What added condi-
tions are needed to permit a quadratic equation to pass through all of the points?
Problem 1.79 Suppose that y D ax
2
C bx C c is a quadratic equation with no roots and that,
when you plug in 2 for x, y D 4. What can we deduce about a?
Problem 1.80 A quadratic equation is said to be a perfect square if it has the form f .x/ D
.x a/
2
for some constant a. What is the discriminant of a perfect square?
Problem 1.81 Find three different quadratic equations with roots 2 and -2.
Problem1.82 Suppose we have a quadratic equation with two roots. Is there another quadratic
equation whose graph has exactly the same shape but that has a root only at zero. Either explain
why not or find an example for y D x
2
3x C 2.
Problem 1.83 Suppose that the graph of two quadratic equations enclose a finite area. What
can you deduce about the equations from this fact?
..................Content has been hidden....................
You can't read the all page of ebook, please click here login for view all page.