If a + b + c = 1, ab + bc + ca = 2 and abc = 3, then the value of a4 + b4 + c4 is equal to ______________.
Answer (integer)
13
Solution
(a + b + c)<sup>2</sup> = 1
<br><br>$\Rightarrow$ a<sup>2</sup>
+ b<sup>2</sup>
+ c<sup>2</sup>
+ 2(ab + bc + ca) = 1
<br><br>$\Rightarrow$ a<sup>2</sup> + b<sup>2</sup>
+ c<sup>2</sup> = – 3 ….(i)
<br><br>$\Rightarrow$ ab + bc + ca = 2 ….(ii)
<br><br>Squaring of equation (ii),
<br><br>$\Rightarrow$ a<sup>2</sup>b<sup>2</sup>
+ b<sup>2</sup>c<sup>2</sup>
+ c<sup>2</sup>a<sup>2</sup>
+ 2(ab<sup>2</sup>c + bc<sup>2</sup>a + ca<sup>2</sup>b) = 4
<br><br>$\Rightarrow$ a<sup>2</sup>b<sup>2</sup>
+ b<sup>2</sup>c<sup>2</sup>
+ c<sup>2</sup>a<sup>2</sup>
+ 2abc(a + b + c) = 4
<br><br>$\Rightarrow$ a<sup>2</sup>b<sup>2</sup>
+ b<sup>2</sup>c<sup>2</sup>
+ c<sup>2</sup>a<sup>2</sup>
+ 6 = 4
<br><br>$\Rightarrow$ a<sup>2</sup>b<sup>2</sup>
+ b<sup>2</sup>c<sup>2</sup>
+ c<sup>2</sup>a<sup>2</sup> = – 2 ….(iii)
<br><br>Squaring of equation (i),
<br><br>$\Rightarrow$ a<sup>4</sup>
+ b<sup>4</sup>
+ c<sup>4</sup>
+ 2(a<sup>2</sup>b<sup>2</sup>
+ b<sup>2</sup>c<sup>2</sup>
+ c<sup>2</sup>a<sup>2</sup>) = 9
<br><br>$\Rightarrow$ a<sup>4</sup>
+ b<sup>4</sup>
+ c<sup>4</sup> – 4 = 9
<br><br>$\Rightarrow$ a<sup>4</sup>
+ b<sup>4</sup>
+ c<sup>4</sup>
= 13
About this question
Subject: Mathematics · Chapter: Complex Numbers and Quadratic Equations · Topic: Complex Numbers and Argand Plane
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