Solving: $x - \sqrt{x+3} - 17 = 0$

Question

$x - \sqrt{x+3} - 17 = 0$

So i tried this:

$-\sqrt{x+3} = 17 - x$

$\sqrt{x+3} = x - 17$

and then again squaring, but i get quadratic equation which i cannot solve

Answer 1

Hint: Add $\sqrt{x + 3}$ to both sides, then square, and manipulate variables. Then use the quadratic formula.

Answer 2

Well, we have:

$$x-\sqrt{x+3}-17=0\tag1$$

Subtract $x-17$ from both sides:

$$-\sqrt{x+3}=17-x\tag2$$

Raise both sides to the power of two:

$$x+3=\left(17-x\right)^2=x^2-34x+289\tag3$$

Subtract $x^2-34 x+289$ from both sides:

$$-x^2+35x-286=-\left(x-22\right)\left(x-13\right)=0\tag4$$

Answer 3

THE PROBLEM of Consstraint

your equation make any sens if $x+3\ge 0 $ so we have
$$x - \sqrt{x+3} - 17 = 0\Longleftrightarrow ~~~0\le \sqrt{x+3} = x - 17~~and~~~ x+3\ge 0$$

that is $$ x - 17\ge 0 ~~and~~~ x+3\ge 0 \Longleftrightarrow x \ge 17~~and~~~ x\ge -3$$ Wich implies that, $x \ge 17$

The on the equation is, $x\ge 17$. Now from this we get

$$(x-17)^2 = x+3 \Longleftrightarrow x^2 -35x +286 = 0$$

$$\Delta = 35^2 - 286 × 4 = 81 =9^2$$ Hence we have $$ x = \frac{35 +9}{2} = 22~~~or~~~x = \frac{35 -9}{2} = 13$$

But dont forget that we have the constraint $x\ge 17$

$$\text{THE ONLY SOLUTION is }~~x = 22$$

Answer 4

It's $$x+3-\sqrt{x+3}-20=0$$ or $$(\sqrt{x+3}-5)(\sqrt{x+3}+4)=0$$ or $$\sqrt{x+3}=5$$ or $$x=22.$$