https://file.aiquickdraw.com/mathbot/file/17254879076693ptborgd.jpg

To solve the equation given in the image, we start by isolating the square root term and then squaring both sides to eliminate the square root. The equation is:

\[
2\sqrt{x} - 2 - 5 = -x
\]

Let's solve it step by step.

### Step 1: Simplify the equation
First, combine the constant terms on the left-hand side:

\[
2\sqrt{x} - 7 = -x
\]

### Step 2: Isolate the square root term
Add \(x\) to both sides of the equation:

\[
2\sqrt{x} = x + 7
\]

### Step 3: Square both sides
Square both sides of the equation to eliminate the square root:

\[
(2\sqrt{x})^2 = (x + 7)^2
\]

\[
4x = x^2 + 14x + 49
\]

### Step 4: Rearrange the equation
Move all terms to one side to set the equation to zero:

\[
x^2 + 14x + 49 - 4x = 0
\]

\[
x^2 + 10x + 49 = 0
\]

### Step 5: Solve the quadratic equation
To solve the quadratic equation \(x^2 + 10x + 49 = 0\), we can use the quadratic formula:

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

Here, \(a = 1\), \(b = 10\), and \(c = 49\). Substituting these values in:

\[
x = \frac{-10 \pm \sqrt{10^2 - 4(1)(49)}}{2(1)}
\]

\[
x = \frac{-10 \pm \sqrt{100 - 196}}{2}
\]

\[
x = \frac{-10 \pm \sqrt{-96}}{2}
\]

\[
x = \frac{-10 \pm 4i\sqrt{6}}{2}
\]

\[
x = -5 \pm 2i\sqrt{6}
\]

### Final Answer
The solutions to the equation are:

\[
x = -5 + 2i\sqrt{6} \quad \text{or} \quad x = -5 - 2i\sqrt{6}
\]

These are complex numbers, so there are no real solutions to the equation.

Would you like more details or have any questions?

---

**Related Questions:**

1. How do you solve equations involving square roots?
2. What is the quadratic formula and when is it used?
3. What are complex numbers and how are they represented?
4. How can we determine if an equation will result in complex solutions?
5. What are the steps to simplify a square root equation?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) when solving quadratic equations to determine if the solutions are real or complex.

Solve the given equation: 2√x - 2 - 5 = -x

Learn how to solve a radical equation 2√x - 7 = -x, involving square roots, quadratic equations, and complex number solutions. Suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation