findi the solutions of each following equation 1. X(x-10) =-21, 2. (x+1)² + (x-3)² =15, 3. 1/3x + 4x/6 =1

Let's solve each of the given equations one by one.

### 1. \( x(x - 10) = -21 \)

This is a quadratic equation. We can expand and then solve it:

\[
x^2 - 10x = -21
\]

Add 21 to both sides:

\[
x^2 - 10x + 21 = 0
\]

Now, we factor the quadratic equation:

\[
(x - 7)(x - 3) = 0
\]

So, the solutions are:

\[
x = 7 \quad \text{or} \quad x = 3
\]

### 2. \( (x + 1)^2 + (x - 3)^2 = 15 \)

First, expand the squares:

\[
(x + 1)^2 = x^2 + 2x + 1
\]
\[
(x - 3)^2 = x^2 - 6x + 9
\]

Now, add these:

\[
x^2 + 2x + 1 + x^2 - 6x + 9 = 15
\]

Simplify the equation:

\[
2x^2 - 4x + 10 = 15
\]

Subtract 15 from both sides:

\[
2x^2 - 4x - 5 = 0
\]

Divide the entire equation by 2:

\[
x^2 - 2x - 2.5 = 0
\]

We can solve this quadratic equation using the quadratic formula:

\[
x = \frac{-(-2) \pm \sqrt{(-2)^2 - 4(1)(-2.5)}}{2(1)}
\]
\[
x = \frac{2 \pm \sqrt{4 + 10}}{2}
\]
\[
x = \frac{2 \pm \sqrt{14}}{2}
\]
\[
x = 1 \pm \frac{\sqrt{14}}{2}
\]

So, the solutions are:

\[
x = 1 + \frac{\sqrt{14}}{2} \quad \text{or} \quad x = 1 - \frac{\sqrt{14}}{2}
\]

### 3. \( \frac{1}{3}x + \frac{4x}{6} = 1 \)

Combine the terms on the left side:

\[
\frac{1}{3}x + \frac{2}{3}x = 1
\]

Simplify:

\[
\frac{3x}{3} = 1
\]

This simplifies further to:

\[
x = 1
\]

### Final Solutions:

1. \( x = 7 \) or \( x = 3 \)
2. \( x = 1 + \frac{\sqrt{14}}{2} \) or \( x = 1 - \frac{\sqrt{14}}{2} \)
3. \( x = 1 \)

Would you like further details or explanations on any of these solutions?

Here are five related questions to consider:
1. How do you derive the quadratic formula?
2. What are some common methods for solving quadratic equations?
3. How do you factor quadratic equations with no real solutions?
4. What are the geometric interpretations of solutions to quadratic equations?
5. How does completing the square help in solving quadratic equations?

**Tip:** When solving quadratic equations, always check if the equation can be easily factored before applying the quadratic formula. This can save time and reduce the risk of errors.

Explore solutions to quadratic equations including x(x - 10) = -21, (x + 1)² + (x - 3)² = 15, and 1/3x + 4x/6 = 1. Learn step-by-step methods and formulas used in solving these equations.

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