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Hey — nice problem choice! Let's work through **Solve for x: 3x² − 12x + 9 = 0** together, step by step. I'll explain every move and keep things simple. You've got this! 😊

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**1) Understand the problem**

We need the value(s) of x that make the equation true:

3x² − 12x + 9 = 0.

This is a quadratic equation (because it has an x² term). For beginner context: quadratics often look like ax² + bx + c = 0. Solutions are the values of x where the parabola crosses the x-axis.

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**2) Simplify if possible**

All terms share a common factor of 3, so divide the whole equation by 3 to make it easier:

(3x²)/3 − (12x)/3 + 9/3 = 0/3

which simplifies to:

x² − 4x + 3 = 0.

Always check for common factors first — it makes life simpler.

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**3) Try factoring (a simple method)**

We look for two numbers that:
• multiply to +3 (the constant term), and
• add to −4 (the coefficient of x).

The numbers −1 and −3 work because:

(−1) × (−3) = 3 and
(−1) + (−3) = −4.

So we can factor:

x² − 4x + 3 = (x − 1)(x − 3).

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**4) Use the Zero Product Property**

If (x − 1)(x − 3) = 0, then either:

x − 1 = 0 ⇒ x = 1, or
x − 3 = 0 ⇒ x = 3.

So the solutions are:

x = 1 or x = 3.

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**5) Check the solutions (always good practice)**

Plug x = 1 into the original equation 3x² − 12x + 9:

3(1)² − 12(1) + 9 = 3 − 12 + 9 = 0 ✓

Plug x = 3:

3(3)² − 12(3) + 9 = 3⋅9 − 36 + 9 = 27 − 36 + 9 = 0 ✓

Both work — nice!

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**Quick summary**

1. Simplify by dividing out common factors → x² − 4x + 3 = 0.
2. Factor into (x − 1)(x − 3).
3. Set each factor to zero → solutions x = 1 and x = 3.
4. Checked both — correct!

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**Extra background (for beginners)**

A quadratic ax² + bx + c = 0 can often be solved by factoring, completing the square, or the quadratic formula:

x = (−b ± √(b² − 4ac)) / (2a).

Factoring is quick when factors are integers. If factoring is hard or impossible with integers, use the quadratic formula.

---

**Practice problems (try these next)**

Work through each using the same steps (simplify → factor if possible → solve → check). I'll give answers so you can check yourself.

**Problem 1:** 2x² − 8x + 6 = 0
→ (Hint: divide by 2 first).
**Answer:** x = 1 or x = 3.

**Problem 2:** x² − 5x + 6 = 0
**Answer:** x = 2 or x = 3.

**Problem 3:** x² + 2x + 1 = 0
**Answer:** x = −1 (a repeated root).

**Problem 4:** 4x² − 4x − 3 = 0
(If factoring is hard, try the quadratic formula).
**Answer:** x = (1 ± √16)/4 = (1 ± 4)/4 → x = 5/4 or x = −3/4.

**Problem 5:** 3x² + 6x + 2 = 0
(Try quadratic formula).
**Answer:** x = (−6 ± 2√3)/6 = −1 ± (√3)/3.

"The Light Beneath the Clouds"

The city floated in the endless white of the upper atmosphere, its shimmering domes reflecting filtered sunlight. The government called it Elystra, the "heaven above storms." To most, it was paradise. To Lira, it was a cage.

Every morning, a mechanical voice echoed through the metallic streets:

"Sunlight allocation begins in five minutes. Citizens, please remain in your designated zones."

Lira watched as the golden panels above the lower levels dimmed once again, the poor districts sinking into shadow. She clenched her fists.

"They're rationing it more every week," she muttered. "Soon, they'll forget the lower rings even exist."

Her younger brother, Jalen, coughed softly behind her. "You shouldn't talk like that, Lira. They'll hear you."

"I don't care," she said, helping him with his oxygen mask. "If I can fix a few drones, I can send more light down there. Maybe even grow some food again."

She wasn't supposed to touch the drones—government property. But every night, she snuck into the maintenance bay with a stolen keycard. Her hands worked quickly, eyes darting between the drone's solar core and the security monitor.

"Come on," she whispered to the little machine. "You're going to shine where they said light doesn't belong."

With a soft click, the drone came to life. Lira smiled and typed a new route: Deliver sunlight to Lower Sector Nine.

**7-Day Meal Plan**

**Day 1**

Breakfast: Greek yogurt with mixed berries and chia seeds

Lunch: Quinoa salad with chickpeas, cucumber, tomatoes, bell peppers, olive oil, and lemon dressing

Snack: Apple slices with sunflower seed butter

Dinner: Lentil and vegetable stir-fry over brown rice

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**Day 2**

Breakfast: Scrambled eggs with spinach and feta cheese

Lunch: Gluten-free wrap with hummus, roasted vegetables, and arugula

Snack: Carrot and celery sticks with tzatziki

Dinner: Baked tofu with roasted broccoli and sweet potatoes

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**Day 3**

Breakfast: Smoothie with spinach, banana, protein powder (plant-based), and almond milk

Lunch: Lentil soup with gluten-free bread

Snack: Handful of roasted pumpkin seeds

Dinner: Stuffed bell peppers with quinoa, black beans, corn, and tomato

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**Day 4**

Breakfast: Overnight oats made with gluten-free oats, chia seeds, and blueberries

Lunch: Chickpea and avocado salad with cherry tomatoes and lemon dressing

Snack: Greek yogurt with sliced strawberries

Dinner: Zucchini noodles with marinara sauce and sautéed tempeh

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**Day 5**

Breakfast: Omelette with mushrooms, bell peppers, and goat cheese

Lunch: Buddha bowl with brown rice, roasted chickpeas, kale, carrots, and tahini dressing

Snack: Orange slices and a handful of sunflower seeds

Dinner: Vegetable curry with lentils served with cauliflower rice

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**Day 6**

Breakfast: Smoothie bowl with spinach, mango, protein powder, and gluten-free granola

Lunch: Quinoa and roasted vegetable salad with feta and olive oil

Snack: Edamame with sea salt

Dinner: Baked tempeh with roasted Brussels sprouts and mashed sweet potatoes

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**Day 7**

Breakfast: Greek yogurt with raspberries and flax seeds

Lunch: Gluten-free wrap with hummus, spinach, roasted zucchini, and bell peppers

Snack: Handful of mixed seeds (pumpkin, sunflower)

Dinner: Chickpea and vegetable stir-fry with brown rice

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**Grocery List** (Approximate Quantities for 7 Days)

**Proteins:**
• Greek yogurt (32 oz)
• Eggs (1 dozen)
• Chickpeas (4 cans or 3 cups cooked)
• Lentils (2 cups dry)
• Tofu (2 blocks)
• Tempeh (2 blocks)
• Protein powder (plant-based, optional)

**Grains & Starches:**
• Quinoa (4 cups cooked)
• Brown rice (3 cups cooked)
• Gluten-free oats (2 cups)
• Gluten-free wraps (5–7)
• Gluten-free bread (1 loaf)

**Vegetables:**
• Spinach (1 large bag)
• Kale (1 bunch)
• Broccoli (2 heads)
• Zucchini (4)
• Bell peppers (6–7)
• Carrots (1 bag)
• Celery (1 bunch)
• Cherry tomatoes (2 cups)
• Sweet potatoes (4–5)
• Mushrooms (1 cup)
• Brussels sprouts (1 lb)

**Fruits:**
• Bananas (5–6)
• Mixed berries (frozen or fresh, 3 cups)
• Strawberries (1 cup)
• Mango (1–2)
• Orange (2–3)
• Apples (3–4)
• Raspberries (1 cup)

**Others:**
• Olive oil
• Sunflower seed butter
• Tahini
• Tzatziki
• Feta or goat cheese (optional)
• Chia seeds
• Flax seeds
• Gluten-free granola
• Lemon (2–3)
• Salt, pepper, spices (cumin, paprika, garlic powder, curry powder, etc.)

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**Prep Instructions & Tips**

• Batch cook grains: Cook all quinoa and brown rice at the beginning of the week and store in the fridge.

• Roast veggies in bulk: Roast bell peppers, zucchini, broccoli, and sweet potatoes with olive oil and spices for 20–25 min at 400°F.

• Prepare proteins ahead: Bake tofu and tempeh with marinade for 20–25 min at 375°F.

• Overnight oats & smoothies: Make overnight oats in jars; smoothies can be prepped in freezer packs.

• Mix salads in jars: Layer ingredients in jars for quick grab-and-go lunches.

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✅ **Tips for variety & nutrition:**

• Rotate vegetables and fruits based on availability and season.

• Use different spices to change flavor profiles (e.g., curry, Italian herbs, smoked paprika).

• Swap quinoa for millet or buckwheat for a different grain option.