Let's solve the quadratic equation:

\[
x^2 - 6x - 10 = -3
\]

### Step 1: Move all terms to one side of the equation

Add \(3\) to both sides of the equation to get:

\[
x^2 - 6x - 10 + 3 = 0
\]

Simplifying, we have:

\[
x^2 - 6x - 7 = 0
\]

### Step 2: Solve using the quadratic formula

The quadratic formula is given by:

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

For the equation \(x^2 - 6x - 7 = 0\), the coefficients are:
- \(a = 1\)
- \(b = -6\)
- \(c = -7\)

Substituting these values into the quadratic formula:

\[
x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(-7)}}{2(1)}
\]

Simplify step by step:

\[
x = \frac{6 \pm \sqrt{36 + 28}}{2}
\]

\[
x = \frac{6 \pm \sqrt{64}}{2}
\]

\[
x = \frac{6 \pm 8}{2}
\]

### Step 3: Find the two possible values for \(x\)

1. \(x = \frac{6 + 8}{2} = \frac{14}{2} = 7\)
2. \(x = \frac{6 - 8}{2} = \frac{-2}{2} = -1\)

### Final Answer:

The solutions are:

\[
x = 7 \quad \text{or} \quad x = -1
\]

Would you like more details or have any questions? Here are 5 related questions you might find interesting:

1. How do you determine whether a quadratic equation has real or complex solutions?
2. What is the significance of the discriminant in a quadratic equation?
3. Can you solve the quadratic equation \(2x^2 + 3x - 2 = 0\) using the quadratic formula?
4. How would you complete the square for the equation \(x^2 - 6x - 7 = 0\)?
5. How do you graph the quadratic function \(y = x^2 - 6x - 7\)?

**Tip:** Always check the discriminant (\(b^2 - 4ac\)) before solving a quadratic equation to know the nature of the roots (real or complex).

Learn how to solve the quadratic equation x^2 - 6x - 10 = -3 with a detailed step-by-step solution. This problem covers key algebraic concepts, including the quadratic formula. Suitable for students in Grades 9-12.

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