Find the value of x x

Algebra Level 2

if log ⁡ 10 ( 98 + x 2 − 12 x + 36 ) = 2 \log_{10}(98+\sqrt{x^{2}-12x+36})=2 , what are the possible values of x x ?

8,4 5,10 6,7 2,9

Jahangir Hossain by Jahangir Hossain , 21, Bangladesh

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2 solutions

Jahangir Hossain
Jan 3, 2017

give that, l o g 10 ( 98 + x 2 − 12 x + 36 ) = 2 log_{10}(98+\sqrt{x^{2}-12x+36})=2

or, 1 0 2 = 98 + x 2 − 12 x + 36 10^{2}=98+\sqrt{x^{2}-12x+36}

or, x 2 − 12 x + 36 = 100 − 98 \sqrt{x^{2}-12x+36}=100-98

or, x 2 − 12 x + 36 = 2 \sqrt{x^{2}-12x+36}=2

or, x 2 − 12 x + 36 = 4 x^{2}-12x+36=4 [squire both sides]

or, x 2 − 12 x + 36 − 4 = 0 x^{2}-12x+36-4=0

or, x 2 − 8 x − 4 x + 32 = 0 x^{2}-8x-4x+32=0

or, x ( x − 8 ) − 4 ( x − 8 ) = 0 x(x-8)-4(x-8)=0

or, ( x − 8 ) ( x − 4 ) = 0 (x-8)(x-4)=0

if, ( x − 8 ) = 0 (x-8)=0

so, x = 8 x=8

Again, ( x − 4 ) = 0 (x-4)=0

so, x = 4 x=4

Answer is 8 , 4 \boxed{8,4}

7 Helpful 0 Interesting 0 Brilliant 0 Confused

Nice Solution.

Shahed Ovi - 4 years, 5 months ago

I believe you need to verify that your solutions satisfy the original equation. Whenever you square both sides, that action introduces another equation with a negative sign in front of the radical. One gets the same solution for both eqns. Fortunately 8 & 4 work for the given eqn., not the other. :)

Richard Costen - 4 years, 5 months ago

3 Helpful 0 Interesting 0 Brilliant 0 Confused

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