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Difference between revisions of "2019 AMC 10A Problems/Problem 19"

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m (Fixed formatting and removed irrelevant attributions)
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==Solution 1==
 
==Solution 1==
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Grouping the first and last terms and two middle terms gives <math>(x^2+5x+4)(x^2+5x+6)+2019</math>, which can be simplified as <math>(x^2+5x+5)^2-1+2019</math>. Squares are nonnegative, (and verifying that <math>x^2+5x+5=0</math> for some <math>x</math>), so the answer is <math>\boxed{\textbf{(B) } 2018}</math>.
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Grouping the first and last terms and two middle terms gives <math>(x^2+5x+4)(x^2+5x+6)+2019</math>, which can be simplified to <math>(x^2+5x+5)^2-1+2019</math>. Noting that squares are nonnegative, and verifying that <math>x^2+5x+5=0</math> for some real <math>x</math>, the answer is <math>\boxed{\textbf{(B) } 2018}</math>.
  
 
==Solution 2==
 
==Solution 2==
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Let <math>a=x+\tfrac{5}{2}</math>. Then <math>(x+1)(x+2)(x+3)(x+4)</math> becomes <math>(a-\tfrac{3}{2})(a-\tfrac{1}{2})(a+\tfrac{1}{2})(a+\tfrac{3}{2})</math>.  
 
Let <math>a=x+\tfrac{5}{2}</math>. Then <math>(x+1)(x+2)(x+3)(x+4)</math> becomes <math>(a-\tfrac{3}{2})(a-\tfrac{1}{2})(a+\tfrac{1}{2})(a+\tfrac{3}{2})</math>.  
  
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We can use difference of squares to get <math>(a^2-\tfrac{9}{4})(a^2-\tfrac{1}{4})</math>, and expand this to get <math>a^4-\tfrac{5}{2}a^2+\tfrac{9}{16}</math>.
+
We can use the difference of two squares to get <math>(a^2-\tfrac{9}{4})(a^2-\tfrac{1}{4})</math>, and expand this to get <math>a^4-\tfrac{5}{2}a^2+\tfrac{9}{16}</math>.
  
 
Refactor this by completing the square to get <math>(a^2-\tfrac{5}{4})^2-1</math>, which has a minimum value of <math>-1</math>. The answer is thus <math>2019-1=\boxed{2018}</math>.  
 
Refactor this by completing the square to get <math>(a^2-\tfrac{5}{4})^2-1</math>, which has a minimum value of <math>-1</math>. The answer is thus <math>2019-1=\boxed{2018}</math>.  
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-WannabeCharmander
 
  
 
==Solution 3 (using calculus)==
 
==Solution 3 (using calculus)==
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To minimize the result, we use <math>y=-5</math>. Hence, the minimum is <math>(-5+4)(-5+6)=-1</math>, so <math>-1+2019 = \boxed{\textbf{(B) }2018}</math>.
 
To minimize the result, we use <math>y=-5</math>. Hence, the minimum is <math>(-5+4)(-5+6)=-1</math>, so <math>-1+2019 = \boxed{\textbf{(B) }2018}</math>.
  
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(inspired by solution by oO8715_alexOo)
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''Note'': The minimum/maximum point of a parabola <math>y = ax^2 + bx + c</math> occurs at <math>x=-\frac{b}{2a}</math>.
  
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Note: The minimum/maximum of a parabola occurs at <math>x=-\frac{b}{2a}</math>.
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==Solution 4==
  
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==Solution 4==
 
 
The expression is negative when an odd number of the factors are negative. This happens when <math>-2 < x < -1</math> or <math>-4 < x -3</math>. Plugging in <math>x = -\frac32</math> or <math>x = -\frac72</math> yields <math>-\frac{15}{16}</math>, which is very close to <math>-1</math>. <math>-1 + 2019 = \boxed{\textbf{(B) }2018}</math>.
 
The expression is negative when an odd number of the factors are negative. This happens when <math>-2 < x < -1</math> or <math>-4 < x -3</math>. Plugging in <math>x = -\frac32</math> or <math>x = -\frac72</math> yields <math>-\frac{15}{16}</math>, which is very close to <math>-1</math>. <math>-1 + 2019 = \boxed{\textbf{(B) }2018}</math>.
  
 
==Solution 5 (using the answer choices) ==
 
==Solution 5 (using the answer choices) ==
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Using the answer choices, we know that <math>\textbf{(C)}</math> , <math>\textbf{(D)}</math> , and <math>\textbf{(E)}</math> are impossible since <math>(x+1)(x+2)(x+3)(x+4)</math> can be negative (as seen when <math>x = -\frac{3}{2}</math>). Plug in <math>x = -\frac{3}{2}</math> to see that it becomes <math>2019 - \frac{15}{16}</math> so round this to <math>\boxed{\textbf{(B) }2018}</math>.
+
 
 +
Using the answer choices, we know that <math>\textbf{(C)}</math> , <math>\textbf{(D)}</math> , and <math>\textbf{(E)}</math> are impossible since <math>(x+1)(x+2)(x+3)(x+4)</math> can be negative (as seen when <math>x = -\frac{3}{2}</math>). Plug in <math>x = -\frac{3}{2}</math> to see that it becomes <math>2019 - \frac{15}{16}</math>, so round this to <math>\boxed{\textbf{(B) }2018}</math>.
  
 
==Video Solution==
 
==Video Solution==

Revision as of 20:37, 17 February 2019

Problem

What is the least possible value of \[(x+1)(x+2)(x+3)(x+4)+2019\]where $x$ is a real number?

$\textbf{(A) } 2017 \qquad\textbf{(B) } 2018 \qquad\textbf{(C) } 2019 \qquad\textbf{(D) } 2020 \qquad\textbf{(E) } 2021$

Solution 1

Grouping the first and last terms and two middle terms gives $(x^2+5x+4)(x^2+5x+6)+2019$, which can be simplified to $(x^2+5x+5)^2-1+2019$. Noting that squares are nonnegative, and verifying that $x^2+5x+5=0$ for some real $x$, the answer is $\boxed{\textbf{(B) } 2018}$.

Solution 2

Let $a=x+\tfrac{5}{2}$. Then $(x+1)(x+2)(x+3)(x+4)$ becomes $(a-\tfrac{3}{2})(a-\tfrac{1}{2})(a+\tfrac{1}{2})(a+\tfrac{3}{2})$.

We can use the difference of two squares to get $(a^2-\tfrac{9}{4})(a^2-\tfrac{1}{4})$, and expand this to get $a^4-\tfrac{5}{2}a^2+\tfrac{9}{16}$.

Refactor this by completing the square to get $(a^2-\tfrac{5}{4})^2-1$, which has a minimum value of $-1$. The answer is thus $2019-1=\boxed{2018}$.

Solution 3 (using calculus)

Similar to Solution 1, grouping the first and last terms and the middle terms, we get $(x^2+5x+4)(x^2+5x+6)+2019$.

Letting $y=x^2+5x$, we get the expression $(y+4)(y+6)+2019$. Now, we can find the critical points of $(y+4)(y+6)$ to minimize the function:

$\frac{d}{dx}(y^2+10y+24)=0$

$2y+10=0$

$2y(y+5)=0$

$y=-5,0$

To minimize the result, we use $y=-5$. Hence, the minimum is $(-5+4)(-5+6)=-1$, so $-1+2019 = \boxed{\textbf{(B) }2018}$.

Note: The minimum/maximum point of a parabola $y = ax^2 + bx + c$ occurs at $x=-\frac{b}{2a}$.

Solution 4

The expression is negative when an odd number of the factors are negative. This happens when $-2 < x < -1$ or $-4 < x -3$. Plugging in $x = -\frac32$ or $x = -\frac72$ yields $-\frac{15}{16}$, which is very close to $-1$. $-1 + 2019 = \boxed{\textbf{(B) }2018}$.

Solution 5 (using the answer choices)

Using the answer choices, we know that $\textbf{(C)}$ , $\textbf{(D)}$ , and $\textbf{(E)}$ are impossible since $(x+1)(x+2)(x+3)(x+4)$ can be negative (as seen when $x = -\frac{3}{2}$). Plug in $x = -\frac{3}{2}$ to see that it becomes $2019 - \frac{15}{16}$, so round this to $\boxed{\textbf{(B) }2018}$.

Video Solution

For those who want a video solution: https://www.youtube.com/watch?v=Mfa7j2BoNjI

See Also

2019 AMC 10A (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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