Difference between revisions of "1983 AIME Problems/Problem 3"
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What is the product of the [[real]] [[root]]s of the [[equation]] <math>x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}</math>? | What is the product of the [[real]] [[root]]s of the [[equation]] <math>x^2 + 18x + 30 = 2 \sqrt{x^2 + 18x + 45}</math>? | ||
− | + | == Solution 1 == | |
− | + | If we were to expand by squaring, we would get a [[quartic Equation|quartic]] [[polynomial]], which isn't always the easiest thing to deal with. | |
− | If we expand by squaring, we get a quartic [[polynomial]], which isn't always the easiest thing to deal with. | ||
− | Instead, we substitute <math>y</math> for <math>x^2+18x+30</math> | + | Instead, we substitute <math>y</math> for <math>x^2+18x+30</math>, so that the equation becomes <math>y=2\sqrt{y+15}</math>. |
− | Now we can square; solving for <math>y</math>, we get <math>y=10</math> or <math>y=-6</math> | + | Now we can square; solving for <math>y</math>, we get <math>y=10</math> or <math>y=-6</math>. The second root is extraneous since <math>2\sqrt{y+15}</math> is always non-negative (and moreover, plugging in <math>y=-6</math>, we get <math>-6=6</math>, which is obviously false). Hence we have <math>y=10</math> as the only solution for <math>y</math>. Substituting <math>x^2+18x+30</math> back in for <math>y</math>, |
− | <center><math>x^2+18x+30=10 \Longrightarrow x^2+18x+20=0.</math></center> | + | <center><math>x^2+18x+30=10 \Longrightarrow x^2+18x+20=0.</math></center> Both of the roots of this equation are real, since its discriminant is <math>18^2 - 4 \cdot 1 \cdot 20 = 244</math>, which is positive. Thus by [[Vieta's formulas]], the product of the real roots is simply <math>\boxed{020}</math>. |
− | + | == Solution 2 == | |
We begin by noticing that the polynomial on the left is <math>15</math> less than the polynomial under the radical sign. Thus: <cmath>(x^2+ 18x + 45) - 2\sqrt{x^2+18x+45} - 15 = 0.</cmath> Letting <math>n = \sqrt{x^2+18x+45}</math>, we have <math>n^2-2n-15 = 0 \Longrightarrow (n-5)(n+3) = 0</math>. Because the square root of a real number can't be negative, the only possible <math>n</math> is <math>5</math>. | We begin by noticing that the polynomial on the left is <math>15</math> less than the polynomial under the radical sign. Thus: <cmath>(x^2+ 18x + 45) - 2\sqrt{x^2+18x+45} - 15 = 0.</cmath> Letting <math>n = \sqrt{x^2+18x+45}</math>, we have <math>n^2-2n-15 = 0 \Longrightarrow (n-5)(n+3) = 0</math>. Because the square root of a real number can't be negative, the only possible <math>n</math> is <math>5</math>. | ||
Substituting that in, we have <cmath>\sqrt{x^2+18x+45} = 5 \Longrightarrow x^2 + 18x + 45 = 25 \Longrightarrow x^2+18x+20=0.</cmath> | Substituting that in, we have <cmath>\sqrt{x^2+18x+45} = 5 \Longrightarrow x^2 + 18x + 45 = 25 \Longrightarrow x^2+18x+20=0.</cmath> | ||
− | + | Reasoning as in Solution 1, the product of the roots is <math>\boxed{020}</math>. | |
+ | |||
+ | == Solution 3 == | ||
+ | |||
+ | Begin by completing the square on both sides of the equation, which gives <cmath>(x+9)^2-51=2\sqrt{(x+3)(x+15)}</cmath> | ||
+ | Now by substituting <math>y=x+9</math>, we get <math>y^2-51=2\sqrt{(y-6)(y+6)}</math>, or <cmath>y^4-106y^2+2745=0</cmath> | ||
+ | The solutions in <math>y</math> are then <cmath>y=x+9=\pm3\sqrt{5},\pm\sqrt{61}</cmath> | ||
+ | Turns out, <math>\pm3\sqrt{5}</math> are a pair of extraneous solutions. Thus, our answer is then <cmath>\left(\sqrt{61}-9\right)\left(-\sqrt{61}-9\right)=81-61=\boxed{020}</cmath> | ||
+ | By difference of squares. | ||
+ | |||
+ | == Solution 4 == | ||
+ | |||
+ | We are given the equation | ||
+ | <cmath>x^2+18x+30=2\sqrt{x^2+18x+45}</cmath> | ||
+ | Squaring both sides yields | ||
+ | <cmath>(x^2+18x+30)^2=4(x^2+18x+45)</cmath> | ||
+ | <cmath>(x^2+18x+30)^2=4(x^2+18x+30+15)</cmath> | ||
+ | <cmath>(x^2+18x+30)^2=4(x^2+18x+30)+60</cmath> | ||
+ | <cmath>(x^2+18x+30)^2-4(x^2+18x+30)-60=0</cmath> | ||
+ | Substituting <math>y=x^2+18x+30</math> yields | ||
+ | <cmath>y^2-4y-60=0</cmath> | ||
+ | <cmath>(y+6)(y-10)=0</cmath> | ||
+ | Thus <math>y=x^2+18x+30=-6,10</math>. However if <math>y=-6</math>, the left side of the equation | ||
+ | <cmath>x^2+18x+30=2\sqrt{x^2+18x+45}</cmath> | ||
+ | would be negative while the right side is negative. Thus <math>y=10</math> is the only possible value and we have | ||
+ | <cmath>x^2+18x+30=10</cmath> | ||
+ | <cmath>x^2+18x+20=0</cmath> | ||
+ | Since the discriminant <math>\sqrt{18^2-4\cdot20}</math> is real, both the roots are real. Thus by Vieta's Formulas, the product of the roots is the constant, <math>\boxed{20}</math>. | ||
+ | |||
+ | ~ Nafer | ||
== See Also == | == See Also == |
Latest revision as of 05:18, 21 July 2022
Problem
What is the product of the real roots of the equation ?
Solution 1
If we were to expand by squaring, we would get a quartic polynomial, which isn't always the easiest thing to deal with.
Instead, we substitute for
, so that the equation becomes
.
Now we can square; solving for , we get
or
. The second root is extraneous since
is always non-negative (and moreover, plugging in
, we get
, which is obviously false). Hence we have
as the only solution for
. Substituting
back in for
,
![$x^2+18x+30=10 \Longrightarrow x^2+18x+20=0.$](http://latex.artofproblemsolving.com/4/f/b/4fbfb8c3979ed14768e003b2256da6a1b2b0fd3e.png)
Both of the roots of this equation are real, since its discriminant is , which is positive. Thus by Vieta's formulas, the product of the real roots is simply
.
Solution 2
We begin by noticing that the polynomial on the left is less than the polynomial under the radical sign. Thus:
Letting
, we have
. Because the square root of a real number can't be negative, the only possible
is
.
Substituting that in, we have
Reasoning as in Solution 1, the product of the roots is .
Solution 3
Begin by completing the square on both sides of the equation, which gives
Now by substituting
, we get
, or
The solutions in
are then
Turns out,
are a pair of extraneous solutions. Thus, our answer is then
By difference of squares.
Solution 4
We are given the equation
Squaring both sides yields
Substituting
yields
Thus
. However if
, the left side of the equation
would be negative while the right side is negative. Thus
is the only possible value and we have
Since the discriminant
is real, both the roots are real. Thus by Vieta's Formulas, the product of the roots is the constant,
.
~ Nafer
See Also
1983 AIME (Problems • Answer Key • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 | ||
All AIME Problems and Solutions |