The discriminant of a quadratic equation worksheet (with solutions ... - Free Printable

You’ve uploaded a worksheet titled “The discriminant of a quadratic equation”. Let’s solve each problem step by step and explain the reasoning.

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## 🔢 Problem 1: Find the value of the discriminant of each quadratic equation.

The discriminant of a quadratic equation in the form
> ax² + bx + c = 0

is given by the formula:

> D = b² – 4ac

It tells us about the nature and number of real solutions:
- If D > 0 → two distinct real solutions
- If D = 0 → one real solution (a repeated root)
- If D < 0 → no real solutions (complex solutions)

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a) 3x² – 5x – 2 = 0

Here,
a = 3, b = –5, c = –2

Discriminant:
D = (–5)² – 4(3)(–2)
= 25 – 4×3×(–2)
= 25 – (–24)
= 25 + 24
= 49

✔ Answer: 49

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b) 3x² – 2x + 1 = 0

Here,
a = 3, b = –2, c = 1

Discriminant:
D = (–2)² – 4(3)(1)
= 4 – 12
= –8

✔ Answer: –8

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## 📊 Problem 2: Determine the number of real solutions of each quadratic equation.

a) x² – 8x + 16 = 0

First, find the discriminant.

a = 1, b = –8, c = 16

D = (–8)² – 4(1)(16)
= 64 – 64
= 0

Since D = 0 → there is exactly one real solution (a repeated root).

✔ Answer: One real solution

*(Note: This equation factors as (x – 4)² = 0, so x = 4 is the double root.)*

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## 📈 Problem 3: Below are the graphs of some parabolas. In each case, determine whether the discriminant is positive, negative, or zero.

*(Note: The image shows two graphs — both appear to be upward-opening parabolas that touch the x-axis at exactly one point — i.e., they are tangent to the x-axis.)*

Key Concept:

- If a parabola crosses the x-axis at two points → D > 0
- If it touches the x-axis at one point → D = 0
- If it never touches the x-axis → D < 0

In both graphs shown (even though only partial images are visible), the parabola appears to touch the x-axis at exactly one point — meaning it has a repeated real root.

✔ So for both graphs, the discriminant is zero.

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## ✔ Final Answers Summary:

Problem 1:

a) Discriminant = 49
b) Discriminant = –8

Problem 2:

a) Number of real solutions = One

Problem 3:

Both graphs → Discriminant = Zero

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Let me know if you’d like to see the solutions graphed or further explained!