Find the value of ab+2cb+3ac?Three positive real numbers a,b,c satisfy the equations a2+3ab+b2=25, b2+c2=9 and a2+ac+c2=16. Then find the value of ab+2cb+3ac?

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High School
Algebra
Algebra I
Quadratic function and equation

Annie Rice

Answered question

2022-04-08

Find the value of

a b + 2 c b + \sqrt{3} a c

?
Three positive real numbers a,b,c satisfy the equations

a^{2} + \sqrt{3} a b + b^{2} = 25, b^{2} + c^{2} = 9 and a^{2} + a c + c^{2} = 16

. Then find the value of

a b + 2 c b + \sqrt{3} a c

Answer & Explanation

titemomo8gjz

Beginner2022-04-09Added 10 answers

The given system is
$a^{2} + \sqrt{3} a b + b^{2} = 25$

$b^{2} + c^{2} = 9$

$a^{2} + a c + c^{2} = 16$
Then $9 + 16 = 25 \Leftrightarrow a^{2} + a c + c^{2} + b^{2} + c^{2} = a^{2} + \sqrt{3} a b + b^{2} \Leftrightarrow c (a + 2 c) = \sqrt{3} a b$
It follows $a b + 2 c b + \sqrt{3} a c = b (a + 2 c) + \sqrt{3} a c = \frac{\sqrt{3} a (b^{2} + c^{2})}{c} = \frac{9 \sqrt{3} a}{c} = X$
Hence one needs to calculate $\frac{a}{c}$ . The successive following steps are clear to understand:
$a^{2} + \sqrt{3} a b + (9 - c^{2}) = 25 \Leftrightarrow b = \frac{16 + c^{2} - a^{2}}{\sqrt{3} a} = \frac{a c + 2 c^{2}}{\sqrt{3} a}$
It follows ${(\frac{a c + 2 c^{2}}{\sqrt{3} a})}^{2} + c^{2} = 9$
So the system $4 c^{4} + 4 a c^{3} + 4 a^{2} c^{2} = 27 a^{2}$

$a^{2} + a c + c^{2} = 16$
Hence $4 c^{2} (16) = 27 a^{2} \Rightarrow \frac{a}{c} = \frac{8}{3 \sqrt{3}}$
Consequently $X = \frac{9 \sqrt{3} \cdot 8}{3 \sqrt{3}} = 24$

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