Absurd

Algebra Level 3

Find the sum of all values of $x$ that satisfy $x^{2}+4x+4x\sqrt{x+3}=13$

The answer is 1.

2 solutions

Rishabh Jain
Jan 17, 2016

Eqn can be written as: $x^{2}+4x\sqrt{x+3}+4(x+3)=25$ $\Rightarrow (x+2\sqrt{x+3})^2=25$ $\Rightarrow (x+2\sqrt{x+3})=\pm5$ $\Rightarrow 4(x+3)=(x\pm5)^2$ $\Rightarrow x^2-14x+13=0 \quad\text{ and }\quad x^2+6x+13=0$ $\color{#D61F06}{x=1,13,-3\pm2i}$ But only x=1 satisfies the given equation, hence only solution x= $\boxed 1$

why does vieta yielded wrong value here as 8?

Chaitnya Shrivastava - 5 years, 4 months ago

That was probably the trickiest part. Since only 1 satisfies the given equation. Other three roots (viz $13,-3\pm2i$ ) are extraneous. Squaring often leads to extraneous solution, we must be careful.

Rishabh Jain - 5 years, 4 months ago

Thank you for the clarification 😊

Chaitnya Shrivastava - 5 years, 4 months ago

@Chaitnya Shrivastava – No problem... 😅

Rishabh Jain - 5 years, 4 months ago

@Rishabh Jain – A little more doubt how can it be determined wether any equation is ordinary or it has something special attached to it I mean where to use vieta's formulae. When the question asks for sum of real values I solve them but when it asks for sum of all values it gets misleading tempting to use the formula so how can we decide about it being ordinary or with extraneous roots.

Chaitnya Shrivastava - 5 years, 4 months ago

@Chaitnya Shrivastava – Gaining experience (Solving more problems). This link might be useful: https://en.m.wikipedia.org/wiki/Extraneous and missing_solutions

If you want to go beyond you must read 'See Also' at the bottom of the above page.

Rishabh Jain - 5 years, 4 months ago

@Rishabh Jain – Thank you !!!!!

Chaitnya Shrivastava - 5 years, 4 months ago

Vijay Kumar
Jan 17, 2016

Since there is an x^2 term in the equation and all the terms in the equation are just added up, so to meet 13, the value of x should be in this set{1,2,3}. Only 1 is the only value of x that satisfies the equation which yields 13. So the answer is just 1 which is the summation of all(just one here)the values of x.

Absurd

The answer is 1.

2 solutions

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