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).