Let's solve the equation:

\[
x^2 - 7 = 7x - x^2
\]

### Step 1: Move all terms to one side of the equation.
First, add \( x^2 \) to both sides to eliminate the \( -x^2 \) on the right-hand side.

\[
x^2 + x^2 - 7 = 7x
\]

This simplifies to:

\[
2x^2 - 7 = 7x
\]

### Step 2: Rearrange the equation.
Move all terms to one side:

\[
2x^2 - 7x - 7 = 0
\]

### Step 3: Solve the quadratic equation using the quadratic formula.
The quadratic formula is:

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

For the equation \( 2x^2 - 7x - 7 = 0 \), the coefficients are:
- \( a = 2 \)
- \( b = -7 \)
- \( c = -7 \)

### Step 4: Calculate the discriminant.
The discriminant is:

\[
\Delta = b^2 - 4ac = (-7)^2 - 4(2)(-7) = 49 + 56 = 105
\]

### Step 5: Find the roots.
Now, substitute the discriminant into the quadratic formula:

\[
x = \frac{-(-7) \pm \sqrt{105}}{2(2)} = \frac{7 \pm \sqrt{105}}{4}
\]

Thus, the two solutions are:

\[
x_1 = \frac{7 + \sqrt{105}}{4} \quad \text{and} \quad x_2 = \frac{7 - \sqrt{105}}{4}
\]

These are the two solutions to the equation.

Would you like further clarification on any steps?

Here are 5 related questions for further exploration:
1. How does the discriminant affect the number of real roots?
2. Can you solve the quadratic equation by completing the square?
3. What is the significance of the quadratic formula in solving equations?
4. How does factoring compare to using the quadratic formula for solving quadratic equations?
5. What are the applications of quadratic equations in real-life scenarios?

**Tip:** Always check your discriminant first—it tells you whether the quadratic has real or complex solutions!

Learn how to solve the quadratic equation X²-7=7x-x² step-by-step using the quadratic formula. This solution explains key algebraic concepts, including rearranging terms, calculating the discriminant, and solving for x. Ideal for students in Grades 8-10.

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