https://file.aiquickdraw.com/mathbot/file/1725858670587nv633fy0.jpg

The given equation from the image is:

\[
x + 2\sqrt{x - 15} = 0
\]

Let's solve this equation.

### Step 1: Isolate the square root term
First, subtract \(x\) from both sides of the equation:

\[
2\sqrt{x - 15} = -x
\]

### Step 2: Square both sides
To eliminate the square root, square both sides of the equation:

\[
(2\sqrt{x - 15})^2 = (-x)^2
\]

Simplifying both sides:

\[
4(x - 15) = x^2
\]

### Step 3: Expand and rearrange the equation
Expand the left-hand side:

\[
4x - 60 = x^2
\]

Next, rearrange the equation to form a quadratic equation:

\[
x^2 - 4x + 60 = 0
\]

### Step 4: Solve the quadratic equation
Now, we'll solve this quadratic equation using the quadratic formula:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Here, \(a = 1\), \(b = -4\), and \(c = 60\). Plugging in these values:

\[
x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(60)}}{2(1)}
\]

Simplifying further:

\[
x = \frac{4 \pm \sqrt{16 - 240}}{2}
\]

\[
x = \frac{4 \pm \sqrt{-224}}{2}
\]

Since the discriminant \(\sqrt{-224}\) is negative, this quadratic equation has no real solutions, only complex solutions. Therefore, the equation \(x + 2\sqrt{x - 15} = 0\) has no real solutions.

Would you like further details on complex solutions or have any questions?

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### Related Questions:
1. How do you solve a quadratic equation when the discriminant is negative?
2. What is the significance of the discriminant in a quadratic equation?
3. How can you graphically represent the equation \(x + 2\sqrt{x - 15} = 0\)?
4. What is the general method to solve equations involving square roots?
5. Can you solve similar equations with different constants, such as \(x + 3\sqrt{x - 20} = 0\)?

### Tip:
Always check the discriminant of a quadratic equation to determine whether it has real or complex solutions.

Learn how to solve square root equations that transform into quadratic equations with this step-by-step solution. The equation x + 2√(x - 15) = 0 is explored, demonstrating concepts such as isolating the square root, squaring both sides, and applying the quadratic formula. Suitable for students in Grades 9-11.

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