To solve the quadratic equation \( x^2 - 20x + 49 = -2 \), we can follow these steps:

### Step 1: Move all terms to one side of the equation.
Add \( 2 \) to both sides:

\[
x^2 - 20x + 49 + 2 = 0
\]

Simplifying:

\[
x^2 - 20x + 51 = 0
\]

### Step 2: Solve the quadratic equation using the quadratic formula.
The quadratic formula is:

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

For the equation \( x^2 - 20x + 51 = 0 \), the coefficients are:
- \( a = 1 \)
- \( b = -20 \)
- \( c = 51 \)

Now, substitute these values into the formula:

\[
x = \frac{-(-20) \pm \sqrt{(-20)^2 - 4(1)(51)}}{2(1)}
\]

Simplifying:

\[
x = \frac{20 \pm \sqrt{400 - 204}}{2}
\]
\[
x = \frac{20 \pm \sqrt{196}}{2}
\]
\[
x = \frac{20 \pm 14}{2}
\]

### Step 3: Find the two possible solutions.
Now, calculate the two possible values for \( x \):

1. \( x = \frac{20 + 14}{2} = \frac{34}{2} = 17 \)
2. \( x = \frac{20 - 14}{2} = \frac{6}{2} = 3 \)

### Final Answer:
The solutions are:
\[
x = 17 \quad \text{or} \quad x = 3
\]

Would you like more details on any part of the process?

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Here are 5 related questions to expand your understanding:
1. How can you factorize a quadratic equation like this one?
2. What is the discriminant, and how does it help in determining the number of real solutions?
3. How does the quadratic formula change when \( b = 0 \)?
4. How can you complete the square to solve this quadratic equation?
5. What are the key differences between solving quadratics by factoring versus using the quadratic formula?

**Tip:** When solving quadratic equations, always check if the discriminant \( b^2 - 4ac \) is positive, zero, or negative to quickly determine the nature of the roots (real or complex).

This webpage offers a step-by-step solution to the quadratic equation x² - 20x + 49 = -2, showing how to use the quadratic formula. Learn the algebraic concepts behind solving quadratic equations, the quadratic formula, and the discriminant. Ideal for students in Grades 9-10.

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