Two irrational equations

Text

Solve the following equations:

• (a) $x-\sqrt{3-x}=-9$
• (b) $\sqrt{25x-x^{2}}-x=3$

Solution

(a) Isolate the root: $\sqrt{3-x}=x+9$. Conditions: $x\le3$ and $x\ge-9$. Squaring: $x^2+19x+78=0\Rightarrow x=-6$ or $x=-13$. Only $x=-6$ satisfies $x\ge-9$; check: $\sqrt{9}=3=x+9$. ✓

(b) $\sqrt{25x-x^2}=x+3$. Conditions: $0\le x\le25$ and $x\ge-3$. Squaring: $2x^2-19x+9=0\Rightarrow x=9$ or $x=\tfrac12$. Both acceptable (checks: $\sqrt{144}=12$, $\sqrt{12{,}25}=3{,}5$). ✓

$\boxed{\text{(a) } x=-6;\qquad \text{(b) } x=\tfrac12 \ \vee\ x=9}$