2019 AMC 8 Problems/Problem 20

Problem 20

How many different real numbers $x$ satisfy the equation $(x^{2}-5)^{2}=16?$

$\textbf{(A) }0\qquad\textbf{(B) }1\qquad\textbf{(C) }2\qquad\textbf{(D) }4\qquad\textbf{(E) }8$

Solution 1

We have that $(x^2-5)^2 = 16$ if and only if $x^2-5 = \pm 4$ . If $x^2-5 = 4$ , then $x^2 = 9 \implies x = \pm 3$ , giving 2 solutions. If $x^2-5 = -4$ , then $x^2 = 1 \implies x = \pm 1$ , giving 2 more solutions. All four of these solutions work, so the answer is $\boxed{\textbf{(D) }4}$ . Further, the equation is a quartic in $x$ , so by the Fundamental Theorem of Algebra, there can be at most four real solutions.

Solution 2

We can expand $(x^2-5)^2$ to get $x^4-10x^2+25$ , so now our equation is $x^4-10x^2+25=16$ . Subtracting $16$ from both sides gives us $x^4-10x^2+9=0$ . Now, we can factor the left hand side to get $(x^2-9)(x^2-1)=0$ . If $x^2-9$ and/or $x^2-1$ equals $0$ , then the whole left side will equal $0$ . Since the solutions can be both positive and negative, we have $4$ solutions: $-3,3,-1,1$ (we can find these solutions by setting $x^2-9$ and $x^2-1$ equal to $0$ and solving for $x$ ). So, the answer is $\boxed{\textbf{(D) }4}$ .

~UnstoppableGoddess

Solution 3

Subtract 16 from both sides and factor using difference of squares:

$(x^2 - 5)^2 = 16$ $(x^2 - 5)^2 - 16 =0$ $(x^2 - 5)^2 - 4^2 = 0$ $[(x^2 - 5)-4][(x^2 - 5) + 4] = 0$ $(x^2 - 9)(x^2 - 1) =0$ $(x+3)(x-3)(x+1)(x-1) = 0$

Quite obviously, this equation has $\boxed{\textbf{(D) }4}$ solutions.

~TaeKim

Solution 3

Associated Video - https://www.youtube.com/watch?v=Q5yfodutpsw

https://youtu.be/0AY1klX3gBo

Solution 4

https://youtu.be/5BXh0JY4klM (Uses a difference of squares & factoring method, different from above solutions)

Solution 5 (video of Solution 1)

https://www.youtube.com/watch?v=44vrsk_CbF8&list=PLLCzevlMcsWNBsdpItBT4r7Pa8cZb6Viu&index=2 ~ MathEx

Video Solution

https://youtu.be/V3HxkJhSn08

-Happytwin

Video Solution

Solution detailing how to solve the problem: https://youtu.be/x4cF3o3Fzj8

Video Solution

https://youtu.be/dh9uf5_ZB5Q

~Education, the Study of Everything

Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)

https://youtu.be/Xm4ZGND9WoY

~Hayabusa1

2019 AMC 8 (Problems • Answer Key • Resources)
Preceded by Problem 19	Followed by Problem 21
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2019 AMC 8 Problems/Problem 20

Contents

Problem 20

Solution 1

Solution 2

Solution 3

Solution 3

Solution 4

Solution 5 (video of Solution 1)

Video Solution

Video Solution

Video Solution

Video Solution by The Power of Logic(Problem 1 to 25 Full Solution)

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