Question

Solve the square root equation −3(2x+3)−−−−−−−−−−√=x
−
3
(
2
𝑥
+
3
)
=
𝑥
.

Answer 1

To solve the equation \(-3\sqrt{2x + 3} = x\), we can start by rearranging the equation to isolate the square root term.

1. Move the square root term to one side:
\[
-3\sqrt{2x + 3} = x
\]

2. Multiply both sides by -1:
\[
3\sqrt{2x + 3} = -x
\]

3. Divide both sides by 3:
\[
\sqrt{2x + 3} = -\frac{x}{3}
\]

4. Square both sides to eliminate the square root:
\[
2x + 3 = \left(-\frac{x}{3}\right)^2
\]

5. Simplify the right side:
\[
2x + 3 = \frac{x^2}{9}
\]

6. Multiply both sides by 9 to eliminate the fraction:
\[
9(2x + 3) = x^2
\]

7. Distribute on the left side:
\[
18x + 27 = x^2
\]

8. Rearrange to form a standard quadratic equation:
\[
x^2 - 18x - 27 = 0
\]

9. Now we can solve this quadratic equation using the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \(a = 1\), \(b = -18\), and \(c = -27\).

10. Calculate the discriminant:
\[
b^2 - 4ac = (-18)^2 - 4(1)(-27) = 324 + 108 = 432
\]

11. Substitute the values into the quadratic formula:
\[
x = \frac{18 \pm \sqrt{432}}{2}
\]

12. Simplify \(\sqrt{432}\):
\[
\sqrt{432} = \sqrt{144 \times 3} = 12\sqrt{3}
\]
Thus,
\[
x = \frac{18 \pm 12\sqrt{3}}{2}
\]
Simplifying this gives:
\[
x = 9 \pm 6\sqrt{3}
\]

So, we have two possible solutions:
\[
x_1 = 9 + 6\sqrt{3}
\]
\[
x_2 = 9 - 6\sqrt{3}
\]

13. Finally, we need to check if these values satisfy the original equation because squaring can introduce extraneous solutions.

For \(x_1 = 9 + 6\sqrt{3}\):
\[
-3\sqrt{2(9 + 6\sqrt{3}) + 3} \stackrel{?}{=} 9 + 6\sqrt{3}
\]
Calculating \(2(9 + 6\sqrt{3}) + 3\):
\[
= 18 + 12\sqrt{3} + 3 = 21 + 12\sqrt{3}
\]
So,
\[
-3\sqrt{21 + 12\sqrt{3}} \stackrel{?}{=} 9 + 6\sqrt{3}
\]
This does not seem to hold.

Now for \(x_2 = 9 - 6\sqrt{3}\):
\[
-3\sqrt{2(9 - 6\sqrt{3}) + 3} \stackrel{?}{=} 9 - 6\sqrt{3}
\]
Calculating \(2(9 - 6\sqrt{3}) + 3\):
\[
= 18 - 12\sqrt{3} + 3 = 21 - 12\sqrt{3}
\]
So,
\[
-3\sqrt{21 - 12\sqrt{3}} \stackrel{?}{=} 9 - 6\sqrt{3}
\]
This also does not seem to hold.

Since neither \(x_1\) nor \(x_2\) satisfy the original equation, we conclude that:

\[
\text{There are no solutions to the equation } -3\sqrt{2x + 3} = x.
\]