We are solving the equation:

$x^2 = -x + 1$

Step 1: Rearranging the terms

Let's move all the terms to one side of the equation:

$x^2 + x - 1 = 0$

Now we have a standard quadratic equation in the form $ax^2 + bx + c = 0$, where:

• $a = 1$
• $b = 1$
• $c = -1$

Step 2: Solving using the quadratic formula

The quadratic formula is given by:

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

Substitute the values of $a$, $b$, and $c$:

$x = \frac{-1 \pm \sqrt{1^2 - 4(1)(-1)}}{2(1)}$

$x = \frac{-1 \pm \sqrt{1 + 4}}{2}$

$x = \frac{-1 \pm \sqrt{5}}{2}$

Step 3: Final Solution

Thus, the solutions are:

$x = \frac{-1 + \sqrt{5}}{2} \quad \text{or} \quad x = \frac{-1 - \sqrt{5}}{2}$

These are the two solutions to the equation.

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Would you like me to clarify any part of the solution? Or do you have another question?

Related Questions:

1. How do we derive the quadratic formula?
2. What happens if the discriminant ($b^2 - 4ac$) is negative?
3. Can you solve quadratic equations using factoring? How?
4. What is the significance of the quadratic formula in real-world problems?
5. What does the graph of a quadratic equation look like?

Tip: Always check the discriminant $b^2 - 4ac$ to determine if the solutions are real or complex!