Let p, q ∈ {1, 2, 3, 4}. The number of equations of the form px² + qx + 1 = 0 having real roots is (

Q: Wait, why do we use $D \ge 0$ instead of $D > 0$?

That is a fantastic question! The condition for 'real roots' includes both distinct real roots ($D > 0$) and equal real roots ($D = 0$). If the problem had asked for 'distinct real roots,' we would strictly use $D > 0$. Since it just says 'real roots,' we must include the case where the roots are identical.

Q: Is there a faster way than checking every case?

For a small set like this, systematic checking is actually the most robust method to avoid errors. However, you can visualize this as finding integer points $(p, q)$ that lie on or above the parabola $q^2 = 4p$ within the $4 \times 4$ square. It is a beautiful intersection of algebra and geometry!

Q: What if $p$ or $q$ were zero? Would the equation still be quadratic?

Great catch! If $p$ were zero, the equation would become $qx + 1 = 0$, which is linear, not quadratic. The problem explicitly defines $p, q \in \{1, 2, 3, 4\}$, so we are safe. Always check the constraints to ensure the definition of the equation holds!

Question

Let $p, q \in \{1, 2, 3, 4\}$. The number of equations of the form $px^2 + qx + 1 = 0$ having real roots is

Options:

(A) 15

(B) 9

(C) 7

(D) 8

Sigma Percentile · Accepted Answer

### The Geometry of Roots: A Journey into the Discriminant

Have you ever looked at a quadratic equation and wondered if it truly 'touches' the x-axis? That is the essence of the discriminant.

Today, we are going to explore the equation $$px^2 + qx + 1 = 0$$, where our coefficients $$p$$ and $$q$$ are restricted to the set $$\{1, 2, 3, 4\}$$. We are on a quest to find how many of these sixteen possible equations yield real roots.

### Phase 1: The Gatekeeper of Reality

Before we dive into the numbers, let us ground ourselves in the theory. For any quadratic equation $$ax^2 + bx + c = 0$$, the roots are determined by the quadratic formula.

The term under the square root, the discriminant $$D = b^2 - 4ac$$, is the gatekeeper. If $$D < 0$$, the roots are imaginary—they exist in a realm beyond the real number line. If $$D \ge 0$$, the roots are real.

Our mission is to count the pairs $$(p, q)$$ that satisfy this condition. Comparing our equation $$px^2 + qx + 1 = 0$$ to the standard form, we identify $$a = p$$, $$b = q$$, and $$c = 1$$.

Substituting these into our discriminant formula, we get $$D = q^2 - 4p$$. Thus, the condition for real roots becomes the master inequality:

$$q^2 \ge 4p$$

### Phase 2: The Systematic Exploration

Imagine a grid where $$p$$ is on the horizontal axis and $$q$$ is on the vertical axis. We have sixteen points to test. Let us proceed row by row, testing each value of $$q$$.

**Case 1: $$q = 1$$**

Substituting $$q = 1$$ into our inequality, we get $$1^2 \ge 4p$$, which simplifies to $$1 \ge 4p$$, or:

$$p \le \frac{1}{4}$$

Since our set for $$p$$ is $$\{1, 2, 3, 4\}$$, there are no values of $$p$$ that satisfy this. The count for this row is zero.

**Case 2: $$q = 2$$**

Now, let $$q = 2$$. Our inequality becomes $$2^2 \ge 4p$$, which is $$4 \ge 4p$$, or $$p \le 1$$.

Only $$p = 1$$ works here. We have found our first valid pair: $$(1, 2)$$.

**Case 3: $$q = 3$$**

With $$q = 3$$, we have $$3^2 \ge 4p$$, or $$9 \ge 4p$$, which means:

$$p \le \frac{9}{4} = 2.25$$

The values $$p = 1$$ and $$p = 2$$ both satisfy this. That gives us two more valid pairs: $$(1, 3)$$ and $$(2, 3)$$.

**Case 4: $$q = 4$$**

Finally, for $$q = 4$$, we get $$4^2 \ge 4p$$, or $$16 \ge 4p$$, which simplifies to $$p \le 4$$.

All four values of $$p$$ ($$1, 2, 3, 4$$) satisfy this condition. This gives us four valid pairs: $$(1, 4), (2, 4), (3, 4), (4, 4)$$.

### Phase 3: The Final Tally

We have methodically traversed the grid. Let us sum our findings: zero from the first row, one from the second, two from the third, and four from the fourth.

$$0 + 1 + 2 + 4 = 7$$

There are exactly **seven** equations that possess real roots. It is a beautiful, clean result, isn't it?

By breaking a seemingly complex problem into a systematic, case-by-case analysis, we have conquered the challenge. Keep this mindset—whenever you face a daunting problem, look for the underlying structure, and build your solution step by step.

Animated Solution for Mathematics - Quadratic Equations: Let $p, q \in {1, 2, 3, 4}$ . The number of equations of the form $p x^{2} + q x + 1 = 0$ having real roots is

Select Answer:

Visualized Solution

Understanding the Equation

Condition for Real Roots

Raw Setup (Substitution)

The Master Inequality

Case 1: $q = 1$

Case 2: $q = 2$

Case 3: $q = 3$

Case 4: $q = 4$

Final Count

The Sigma Insight: Nature of Roots

The Geometry of Roots

Phase 1

Phase 2

Phase 3

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Animated Solution for Mathematics - Quadratic Equations: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Let p,q∈{1,2,3,4}. The number of equations of the form px2+qx+1=0 having real roots is

Select Answer:

Visualized Solution

Understanding the Equation

Condition for Real Roots

Raw Setup (Substitution)

The Master Inequality

Case 1: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } q=1

Case 2: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } q=2

Case 3: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } q=3

Case 4: .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } q=4