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Difference between revisions of "2018 AMC 10A Problems/Problem 10"
Line 41: Line 41:
 
Notice that these two expressions are conjugates of one another. Therefore, we can find that by multiply these two conjugates by one another we should be able to find that:
 
Notice that these two expressions are conjugates of one another. Therefore, we can find that by multiply these two conjugates by one another we should be able to find that:
 
                         Difference of Squares Formula: <math>(a+b)(a-b)=a^2-b^2</math>
 
                         Difference of Squares Formula: <math>(a+b)(a-b)=a^2-b^2</math>
−
                         <math>(\sqrt(49-x^2) - \sqrt(25-x^2))(\sqrt(49-x^2) + \sqrt(25-x^2)) = (49-x^2) - (25-x^2) = 49-x^2-25+x^2 = 24</math>
+
                         <math>(\sqrt(49-x^2) - \sqrt(25-x^2))(\sqrt(49-x^2) + \sqrt(25-x^2)) = (49-x^2) - (25-x^2) </math>
 +
                        <math>\Rightarrow 49-x^2-25+x^2 = 24</math>
 
We are already given that the first expression equals 3, thus, our expression now becomes:
 
We are already given that the first expression equals 3, thus, our expression now becomes:
−
                         <math>3(\sqrt(49-x^2)+\sqrt(25-x^2)) = 24 \Rightarrow \sqrt(49-x^2)+\sqrt(25-x^2) = 8 </math>
+
                         <math>3(\sqrt(49-x^2)+\sqrt(25-x^2)) = 24 </math>
 +
                        <math>\Rightarrow \sqrt(49-x^2)+\sqrt(25-x^2) = 8 </math>
 
Thus, the answer is <math>\boxed{\textbf{(A) }8}</math>
 
Thus, the answer is <math>\boxed{\textbf{(A) }8}</math>
  

Revision as of 22:28, 24 August 2024

Problem

Suppose that real number $x$ satisfies \[\sqrt{49-x^2}-\sqrt{25-x^2}=3\]What is the value of $\sqrt{49-x^2}+\sqrt{25-x^2}$?

$\textbf{(A) }8\qquad \textbf{(B) }\sqrt{33}+8\qquad \textbf{(C) }9\qquad \textbf{(D) }2\sqrt{10}+4\qquad \textbf{(E) }12\qquad$

Solution 1

We let $a=\sqrt{49-x^2}+\sqrt{25-x^2}$; in other words, we want to find $a$. We know that $a\cdot3=\left(\sqrt{49-x^2}+\sqrt{25-x^2}\right)\cdot\left(\sqrt{49-x^2}-\sqrt{25-x^2}\right)=\left(\sqrt{49-x^2}\right)^2-\left(\sqrt{25-x^2}\right)^2=\left(49-x^2\right)-\left(25-x^2\right)=24.$ Thus, $a=\boxed{8}$.

~Technodoggo

Solution 2

Let $a = \sqrt{49-x^2}$, and $b = \sqrt{25-x^2}$. Solving for the constants in terms of x, a , and b, we get $a^2 + x^2 = 49$, and $b^2 + x^2 = 25$. Subtracting the second equation from the first gives us $a^2 - b^2 = 24$. Difference of squares gives us $(a+b)(a-b) = 24$. Since we want to find $a+b = \sqrt{49-x^2}+\sqrt{25-x^2}$, and we know $a-b = 3$, we get $3(a+b) = 24$, so $a+b = \boxed{\textbf{(A) }8}$


~idk12345678

Solution 3

We can substitute $25 - x^2$ for $a$, thus turning the equation into $\sqrt{a+24} - \sqrt{a} = 3$. Moving the $\sqrt{a}$ to the other side and squaring gives us $a + 24 = 9 + 6\sqrt{a} + a$, solving for $a$ gives us 25/4. We substitute this value into the expression they asked us to evaluate giving 8.

~ SAMANTAP

Solution 4

Move $-\sqrt{25-x^2}$ to the right to get $\sqrt{49-x^2} = 3 + \sqrt{25-x^2}$. Square both sides to get $49-x^2 = 9 + 6\sqrt{25-x^2} + (25-x^2)$. Simplify to get $15 = 6\sqrt{25-x^2}$, or $\frac{5}{2} = \sqrt{25-x^2}$ Substitute this back into the original equation tog et that $\sqrt{49-x^2} = \frac{11}{2}$. The answer is $\boxed{\textbf{(A) }8}$

==Solution 5(Jaideep's Difference of Roots Equals Integer Method)[JDRIM] We are given that, 

                        $\sqrt(49-x^2) - \sqrt(25-x^2) = 3$

We are asked to find, 

                        $\sqrt(49-x^2) + \sqrt(25-x^2)$

Notice that these two expressions are conjugates of one another. Therefore, we can find that by multiply these two conjugates by one another we should be able to find that: 

                        Difference of Squares Formula: $(a+b)(a-b)=a^2-b^2$
                        $(\sqrt(49-x^2) - \sqrt(25-x^2))(\sqrt(49-x^2) + \sqrt(25-x^2)) = (49-x^2) - (25-x^2)$
                        $\Rightarrow 49-x^2-25+x^2 = 24$

We are already given that the first expression equals 3, thus, our expression now becomes: 

                        $3(\sqrt(49-x^2)+\sqrt(25-x^2)) = 24$
                        $\Rightarrow \sqrt(49-x^2)+\sqrt(25-x^2) = 8$

Thus, the answer is $\boxed{\textbf{(A) }8}$

~im_space_cadet


~Failure.net

Video Solution (HOW TO THINK CREATIVELY!)

https://youtu.be/P-atxiiTw2I

~Education, the Study of Everything



Video Solutions

Video Solution 1

https://youtu.be/ba6w1OhXqOQ?t=1403

~ pi_is_3.14

Video Solution 2

https://youtu.be/zQG70XKAdeA ~ North America Math Contest Go Go Go

Video Solution 3

https://youtu.be/ZiZVIMmo260

Video Solution 4

https://youtu.be/5cA87rbzFdw

~savannahsolver

See Also

2018 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 9 		Followed by
Problem 11
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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