Visual guide to understanding quadratic roots, featuring graphs for two real roots, one real root, and no real roots.

Absolutely! Let’s fill in the blanks and solve each quadratic equation step by step.

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## ✔ First, fill in the definitions:

parabola – the U-shaped graph of a quadratic function
*(Note: Sometimes called “cup-shaped” or “bell-shaped,” but “U-shaped” is standard for upward-opening parabolas. Downward ones are still U-shaped, just inverted.)*

roots – where the parabola crosses the x-axis
*(Also called zeros, x-intercepts, solutions — all correct!)*

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## 🔍 Now, let’s solve each quadratic to find its roots (zeros):

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🟢 TWO REAL ROOTS

Equation:
x² + 2x − 3 = 0

We can factor this!

Look for two numbers that multiply to -3 and add to +2:
→ +3 and -1

So:
(x + 3)(x − 1) = 0

Set each factor equal to zero:

- x + 3 = 0 → x = -3
- x − 1 = 0 → x = 1

✔ Answer: Two real roots: x = -3 and x = 1

*(This matches the graph — it crosses the x-axis at two points.)*

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🟡 ONE REAL ROOT

Equation:
x² + 6x + 9 = 0

This is a perfect square trinomial!

Notice:
x² + 6x + 9 = (x + 3)²

So:
(x + 3)² = 0

Take square root of both sides:
x + 3 = 0 → x = -3

✔ Answer: One real root: x = -3 *(a repeated root)*

*(This matches the graph — it touches the x-axis at one point, the vertex.)*

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🔴 NO REAL ROOTS

Equation:
(x − 1)² + 3 = 0

Solve step-by-step:

Subtract 3 from both sides:
(x − 1)² = -3

Take square root of both sides:
x − 1 = ±√(-3)

But √(-3) is not a real number — it’s imaginary (√(-3) = i√3)

✔ Answer: No real roots — only complex/imaginary roots.

*(This matches the graph — the parabola never touches or crosses the x-axis.)*

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## 📝 Final Answers to Fill In:

Definitions:

- parabola – the U-shaped graph of a quadratic function
- roots – where the parabola crosses the x-axis

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TWO REAL ROOTS:

> Solve for the zeros:
> x² + 2x − 3 = 0
> → x = -3 and x = 1

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ONE REAL ROOT:

> Solve for the zeros:
> x² + 6x + 9 = 0
> → x = -3

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NO REAL ROOTS:

> Solve for the zeros:
> (x − 1)² + 3 = 0
> → No real solutions (or “no real roots”)

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## 💡 Bonus Tip:
You can also use the discriminant (b² − 4ac) to predict the number of real roots without solving:

- If discriminant > 0 → Two real roots
- If discriminant = 0 → One real root
- If discriminant < 0 → No real roots

Let’s check:

1. x² + 2x − 3 → a=1, b=2, c=-3 → D = 4 + 12 = 16 > 0 → Two real roots ✔
2. x² + 6x + 9 → a=1, b=6, c=9 → D = 36 − 36 = 0 → One real root ✔
3. (x−1)² + 3 = x² − 2x + 1 + 3 = x² − 2x + 4 → a=1, b=-2, c=4 → D = 4 − 16 = -12 < 0 → No real roots ✔

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✔ All done! You now understand how to classify and solve quadratic equations based on their roots. Great job!