Solve the radical equation: −√𝒙 + 𝟐 + 𝒙 = −4

We are tasked with solving the equation:

\[
-\sqrt{x} + 2 + x = -4
\]

### Step 1: Isolate the square root term

Move all the terms except for the square root to one side of the equation:

\[
-\sqrt{x} = -4 - 2 - x
\]

Simplify the right-hand side:

\[
-\sqrt{x} = -6 - x
\]

### Step 2: Eliminate the negative signs

Multiply both sides of the equation by \(-1\) to eliminate the negative square root:

\[
\sqrt{x} = 6 + x
\]

### Step 3: Square both sides

To eliminate the square root, square both sides of the equation:

\[
(\sqrt{x})^2 = (6 + x)^2
\]

This simplifies to:

\[
x = (6 + x)^2
\]

Now, expand the right-hand side:

\[
x = 36 + 12x + x^2
\]

### Step 4: Rearrange into a quadratic equation

Move all terms to one side of the equation:

\[
0 = x^2 + 12x + 36 - x
\]

Simplify:

\[
0 = x^2 + 11x + 36
\]

### Step 5: Solve the quadratic equation

We can solve this quadratic equation using the quadratic formula:

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

For the equation \(x^2 + 11x + 36 = 0\), the coefficients are:
- \(a = 1\)
- \(b = 11\)
- \(c = 36\)

Substitute these values into the quadratic formula:

\[
x = \frac{-11 \pm \sqrt{11^2 - 4(1)(36)}}{2(1)}
\]

Simplify the discriminant:

\[
x = \frac{-11 \pm \sqrt{121 - 144}}{2}
\]
\[
x = \frac{-11 \pm \sqrt{-23}}{2}
\]

### Step 6: Interpret the result

The discriminant \(\sqrt{-23}\) is imaginary, meaning there are no real solutions to this equation.

### Final answer:
The equation has no real solutions.

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Would you like more details or clarifications on this solution?

Here are 5 related questions for practice:
1. Solve the equation \(\sqrt{x} + x = 6\).
2. Solve \(\sqrt{2x + 3} = x - 1\).
3. Find the solution of \(x^2 + 5x + 6 = 0\).
4. Solve the radical equation \(\sqrt{x + 4} = x - 2\).
5. Solve \(x^2 - 8x + 15 = 0\).

**Tip:** Always check for extraneous solutions when solving radical equations by substituting the solutions back into the original equation.

This page provides a detailed step-by-step guide to solving the radical equation -√x + 2 + x = -4. It covers key concepts like isolating the square root, squaring both sides, and solving a quadratic equation using the quadratic formula. Suitable for students in Grades 10-12.

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