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Difficulty: MediumSurds and Indices

If the equation x+x−9=9\sqrt{x} + \sqrt{x - 9} = 9 holds true, what is the exact value of xx?

Answer: 25

Answer

25
The exact value of xx is 25. By moving one radical to the right side and squaring both sides, we eliminate one square root. Simplifying and isolating the remaining square root allows us to square both sides a second time, revealing the final value. Alternatively, using the conjugate property of surds: multiplying both sides of the identity (x)2−(x−9)2=9(\sqrt{x})^2 - (\sqrt{x-9})^2 = 9 by their difference gives (x−x−9)(x+x−9)=9(\sqrt{x} - \sqrt{x-9})(\sqrt{x} + \sqrt{x-9}) = 9. Since the sum is 9, the difference must be 1 (i.e., x−x−9=1\sqrt{x} - \sqrt{x-9} = 1). Adding this back to the original equation yields 2x=102\sqrt{x} = 10, so x=5\sqrt{x} = 5 and x=25x = 25.

Step-by-Step Solution

1
Isolate one of the square root terms on one side of the equation.
x=9−x−9\sqrt{x} = 9 - \sqrt{x - 9}
Isolating a radical makes it easier to eliminate it by squaring both sides.
2
Square both sides of the equation and expand the right side.
x=81−18x−9+(x−9)x = 81 - 18\sqrt{x - 9} + (x - 9)
Squaring eliminates the isolated radical. The right side is expanded using the algebraic identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2.
3
Simplify the equation by canceling xx from both sides and combining constant terms.
x=x+72−18x−9⇒18x−9=72x = x + 72 - 18\sqrt{x - 9} \Rightarrow 18\sqrt{x - 9} = 72
Combining like terms simplifies the equation, leaving only a single radical expression.
4
Divide by 18 and square both sides one final time to solve for xx.
x−9=4⇒x−9=16⇒x=25\sqrt{x - 9} = 4 \Rightarrow x - 9 = 16 \Rightarrow x = 25
Isolating the final radical and squaring removes the remaining root, yielding a simple linear equation for xx.

Key Concept

Solving radical equations and applying algebraic identities with surds.
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Topics

Surds and Indices

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