AMC 8 2019 Problem 20 | Fundamental Theorem of Algebra

Try out this beautiful algebra problem number 2 from AMC 8 2019 based on the Fundamental Theorem of Algebra.

AMC 8 2019 Problem 20:

How many different real numbers $<img loading=$ " width="10" height="8"> satisfy the equation $<p class=$ $(x^{2}-5)^{2}=16?$

" width="119" height="19">

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

Key Concepts

Algebra

Value

Telescoping

Check the Answer

Answer: is (D) 4

AMC 8, 2019, Problem 20

Try with Hints

The given equation is

$<img loading=$ " width="129" height="22"/>

and that means

$<img loading=$ " width="125" height="20"/>

Among both cases, if

$<img loading=$ " width="106" height="20"/>

then,

$<img loading=$ " width="204" height="20"/>

and that means we have 2 different real numbers that satisfy the equation.

and if we take another case, then

$<img loading=$ " width="125" height="20"/>

and so,

$<img loading=$ " width="203" height="20"/>

and that means we have 2 different real numbers in this `case too that satisfy the equation. So total 2+2=4 real numbers that satisfy the equation.

Other useful links

Cheenta Numerates Program for AMC - AIME

Try out this beautiful algebra problem number 2 from AMC 8 2019 based on the Fundamental Theorem of Algebra.

AMC 8 2019 Problem 20:

How many different real numbers $<img loading=$ " width="10" height="8"> satisfy the equation $<p class=$ $(x^{2}-5)^{2}=16?$

" width="119" height="19">

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

Key Concepts

Algebra

Value

Telescoping

Check the Answer

Answer: is (D) 4

AMC 8, 2019, Problem 20

Try with Hints

The given equation is

$<img loading=$ " width="129" height="22"/>

and that means

$<img loading=$ " width="125" height="20"/>

Among both cases, if

$<img loading=$ " width="106" height="20"/>

then,

$<img loading=$ " width="204" height="20"/>

and that means we have 2 different real numbers that satisfy the equation.

and if we take another case, then

$<img loading=$ " width="125" height="20"/>

and so,

$<img loading=$ " width="203" height="20"/>

and that means we have 2 different real numbers in this `case too that satisfy the equation. So total 2+2=4 real numbers that satisfy the equation.

Other useful links

Cheenta Numerates Program for AMC - AIME

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