Difference between revisions of "2013 AMC 10B Problems/Problem 19"
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==Problem== | ==Problem== | ||
− | The real numbers <math>c,b,a</math> form an arithmetic sequence with <math>a\ | + | The real numbers <math>c,b,a</math> form an arithmetic sequence with <math>a \geq b \geq c \geq 0</math>. The quadratic <math>ax^2+bx+c</math> has exactly one root. What is this root? |
<math> \textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} </math> | <math> \textbf{(A)}\ -7-4\sqrt{3}\qquad\textbf{(B)}\ -2-\sqrt{3}\qquad\textbf{(C)}\ -1\qquad\textbf{(D)}\ -2+\sqrt{3}\qquad\textbf{(E)}\ -7+4\sqrt{3} </math> | ||
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==Solutions== | ==Solutions== | ||
===Solution 1=== | ===Solution 1=== | ||
− | It is given that <math>ax^2+bx+c=0</math> has 1 real root, so the discriminant is zero, or <math>b^2=4ac</math>. Because a, b, c are in arithmetic progression, <math>b-a=c-b</math>, or <math>b=\frac {a+c} {2} </math>. We need to find the unique root, or <math>-\frac {b} {2a} </math> (discriminant is 0). From <math>b^2=4ac</math>, we can get <math>-\frac {b} {2a} =-\frac {2c} {b}</math>. Ignoring the negatives, we have | + | It is given that <math>ax^2+bx+c=0</math> has 1 real root, so the discriminant is zero, or <math>b^2=4ac</math>. |
− | <math>\frac {2c} {b} = \frac {2c} {\frac {a+c} {2}} = \frac {4c} {a+c} = \frac {1} {\frac {1} {\frac {4c} {a+c}}} = \frac {1} {\frac {a+c} {4c}} = \frac {1} {\frac {a} {4c} + \frac {1} {4} }</math>. Fortunately, finding <math>\frac {a} {c} </math> is not very hard. Plug in <math>b=\frac {a+c} {2}</math> to <math>b^2=4ac</math>, we have <math>a^2+2ac+c^2=16ac</math>, or <math>a^2-14ac+c^2=0</math>, and dividing by <math>c^2</math> gives <math>(\frac {a} {c} ) ^2-14( \frac {a} {c} ) +1 = 0</math>, so <math>\frac {a} {c} = \frac {14 \pm \sqrt {192} } {2} = 7 \pm 4 \sqrt {3} </math>. But <math>7-4\sqrt {3} <1</math>, violating the assumption that <math>a \ge c</math>. Therefore, <math>\frac {a} {c} = 7 +4\sqrt {3} </math>. Plugging this in, we have <math>\frac {1} {\frac {a} {4c} + \frac {1} {4} } = \frac {1} {2+ \sqrt {3} } = 2- \sqrt {3} </math>. But we need the negative of this, so the answer is <math>\boxed {\textbf{(D)}}.</math> | + | |
+ | Because a, b, c are in arithmetic progression, <math>b-a=c-b</math>, or <math>b=\frac {a+c} {2} </math>. | ||
+ | |||
+ | We need to find the unique root, or <math>-\frac {b} {2a} </math> (discriminant is 0). | ||
+ | |||
+ | From <math>b^2=4ac</math>, we can get <math>-\frac {b} {2a} =-\frac {2c} {b}</math>. | ||
+ | |||
+ | Ignoring the negatives(for now), we have | ||
+ | <math>\frac {2c} {b} = \frac {2c} {\frac {a+c} {2}} = \frac {4c} {a+c} = \frac {1} {\frac {1} {\frac {4c} {a+c}}} = \frac {1} {\frac {a+c} {4c}} = \frac {1} {\frac {a} {4c} + \frac {1} {4} }</math>. | ||
+ | |||
+ | Fortunately, finding <math>\frac {a} {c} </math> is not very hard. | ||
+ | |||
+ | Plug in <math>b=\frac {a+c} {2}</math> to <math>b^2=4ac</math>, we have <math>a^2+2ac+c^2=16ac</math>, or <math>a^2-14ac+c^2=0</math>, and dividing by <math>c^2</math> gives <math>\left( \frac {a} {c} \right) ^2-14 \left( \frac {a} {c} \right) +1 = 0</math>, so <math>\frac {a} {c} = \frac {14 \pm \sqrt {192} } {2} = 7 \pm 4 \sqrt {3} </math>. | ||
+ | |||
+ | But <math>7-4\sqrt {3} <1</math>, violating the assumption that <math>a \ge c</math>. | ||
+ | |||
+ | Therefore, <math>\frac {a} {c} = 7 +4\sqrt {3} </math>. | ||
+ | |||
+ | Plugging this in, we have <math>\frac {1} {\frac {a} {4c} + \frac {1} {4} } = \frac {1} {2+ \sqrt {3} } = 2- \sqrt {3} </math>. | ||
+ | |||
+ | But we need the negative of this, so the answer is <math>\boxed {\textbf{(D)}}.</math> | ||
===Solution 2=== | ===Solution 2=== | ||
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<math>(x-r)^2 = x^2 - 2rx + r^2</math> where <math>1 \ge -2r \ge r^2 \ge 0</math>. | <math>(x-r)^2 = x^2 - 2rx + r^2</math> where <math>1 \ge -2r \ge r^2 \ge 0</math>. | ||
We now use the fact that the coefficients are in an arithmetic sequence. Note that in any arithmetic sequence, the average is equal to the median. Thus, | We now use the fact that the coefficients are in an arithmetic sequence. Note that in any arithmetic sequence, the average is equal to the median. Thus, | ||
− | <math>r^2 + 1 = -4r</math> and <math>r = -2 \pm \sqrt{3}</math>. Since <math>1 | + | <math>r^2 + 1 = -4r</math> and <math>r = -2 \pm \sqrt{3}</math>. Since <math>1 \ge r^2</math>, we easily see that <math>|r|</math> has to be between 1 and 0. Thus, we can eliminate <math>-2 - \sqrt{3}</math> and are left with <math>\boxed{\textbf{(D)} -2 + \sqrt{3}}</math> as the answer. |
===Solution 3=== | ===Solution 3=== | ||
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Setting the two equations equal, we have <math>4ac=\frac{a^2+2ac+c^2}{4}</math>. | Setting the two equations equal, we have <math>4ac=\frac{a^2+2ac+c^2}{4}</math>. | ||
− | <math>\implies 16ac=a^2+2ac+c^2\implies 14ac=a^2+c^2\implies a^2-14ac+c^2</math>. | + | <math>\implies 16ac=a^2+2ac+c^2\implies 14ac=a^2+c^2\implies a^2-14ac+c^2=0</math>. |
Dividing by <math>c^2</math>, we get <math>(\frac{a}{c})^2-14(\frac{a}{c})+1=0</math>. | Dividing by <math>c^2</math>, we get <math>(\frac{a}{c})^2-14(\frac{a}{c})+1=0</math>. | ||
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Hence, <math>r^2=7-4\sqrt3</math>. Solving for <math>r</math>, we have <math>r</math> equals <math>-2+\sqrt3=\boxed{D}</math>. | Hence, <math>r^2=7-4\sqrt3</math>. Solving for <math>r</math>, we have <math>r</math> equals <math>-2+\sqrt3=\boxed{D}</math>. | ||
+ | |||
+ | ===Solution 5=== | ||
+ | |||
+ | The unique root is <math>-\frac {b} {2a} </math> as discriminant has to be 0. | ||
+ | |||
+ | We have <math>a-b=b-c \implies c=2b-a</math>. | ||
+ | |||
+ | Also discriminant <math>b^2-4ac=0 \implies b^2=4ac</math>. Plug in <math>c=2b-a</math> from above, we have <math>b^2=4a(2b-a) \implies b^2-8ab+4a^2=0</math>. Divide both sides by <math>a^2</math>, we have <math>(\frac{b}{a})^2-8(\frac{b}{a})+4=0</math>. | ||
+ | |||
+ | Solve <math>\frac{b}{a}</math>, we get <math>4\pm2\sqrt {3}</math>. Since <math>a \ge b</math>, <math>\frac{b}{a}=4-2\sqrt {3}</math>. | ||
+ | |||
+ | Therefore the unique root <math>-\frac{b}{2a} = -2+\sqrt3=\boxed{D}</math>. | ||
+ | |||
+ | ===Solution 6 (guess)=== | ||
+ | |||
+ | We can just try out the answer choices! First, (C) clearly fails. Then, we try (B). It fails too. But <math>\boxed{D}</math> works. | ||
+ | |||
+ | *feel free to add details | ||
+ | |||
+ | ~Yrock | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/jCxm08uuW7Q | ||
+ | |||
+ | ~savannahsolver | ||
== See also == | == See also == | ||
{{AMC10 box|year=2013|ab=B|num-b=18|num-a=20}} | {{AMC10 box|year=2013|ab=B|num-b=18|num-a=20}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 11:49, 1 June 2024
Contents
Problem
The real numbers form an arithmetic sequence with
. The quadratic
has exactly one root. What is this root?
Solutions
Solution 1
It is given that has 1 real root, so the discriminant is zero, or
.
Because a, b, c are in arithmetic progression, , or
.
We need to find the unique root, or (discriminant is 0).
From , we can get
.
Ignoring the negatives(for now), we have
.
Fortunately, finding is not very hard.
Plug in to
, we have
, or
, and dividing by
gives
, so
.
But , violating the assumption that
.
Therefore, .
Plugging this in, we have .
But we need the negative of this, so the answer is
Solution 2
Note that we can divide the polynomial by to make the leading coefficient 1 since dividing does not change the roots or the fact that the coefficients are in an arithmetic sequence. Also, we know that there is exactly one root so this equation must be of the form
where
.
We now use the fact that the coefficients are in an arithmetic sequence. Note that in any arithmetic sequence, the average is equal to the median. Thus,
and
. Since
, we easily see that
has to be between 1 and 0. Thus, we can eliminate
and are left with
as the answer.
Solution 3
Given that has only 1 real root, we know that the discriminant must equal 0, or that
. Because the discriminant equals 0, we have that the root of the quadratic is
. We are also given that the coefficients of the quadratic are in arithmetic progression, where
. Letting the arbitrary difference equal variable
, we have that
and that
. Plugging those two equations into
, we have
which yields
. Isolating
, we have
. Substituting that in for
in
, we get
. Once again, substituting that in for
in
, we have
. The answer is:
Solution 4
We have
Hence
.
And we have .
Squaring the first expression, and dividing by 4, we get
.
Setting the two equations equal, we have .
.
Dividing by , we get
.
Setting we have
Solving for , we get
We know that
, so
.
By Vieta's, we know that , where
equals the double root of the quadratic.
So, we get
. After rationalizing the denominator, we get
.
Hence, . Solving for
, we have
equals
.
Solution 5
The unique root is as discriminant has to be 0.
We have .
Also discriminant . Plug in
from above, we have
. Divide both sides by
, we have
.
Solve , we get
. Since
,
.
Therefore the unique root .
Solution 6 (guess)
We can just try out the answer choices! First, (C) clearly fails. Then, we try (B). It fails too. But works.
- feel free to add details
~Yrock
Video Solution
~savannahsolver
See also
2013 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AMC 10 Problems and Solutions |
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