Since real roots...∆>=0
so here ∆=q^2-4p
since discriminant is greater than or equal to zero
=>..... q^2-4p>=0
=>....q^2>=4p
=>.....now the equation satisfies... when p=1and q=2
we can consider.. when p=1 then q={2,3,4}
when p=2 then q={3,4}
when p=3 then q={4}
when p=4 then q={4}
so the solutions (p q) are (1 2) (1 3) (1 4) (2 3) (2 4) (3 4) (4 4).....total 7

Number of Equations with Real Roots

To find the number of equations of the form px^2 + qx + 1 with real roots, we need to analyze the discriminant of the quadratic equation.

Quadratic Equation
A quadratic equation has the general form ax^2 + bx + c = 0, where a, b, and c are constants.

Discriminant
The discriminant of a quadratic equation is given by the formula Δ = b^2 - 4ac. It determines the nature of the roots of the equation.

Real Roots
For a quadratic equation to have real roots, the discriminant must be greater than or equal to zero, i.e., Δ ≥ 0.

Analysis

Let's consider each value of p and q from the set {1, 2, 3, 4} and analyze the discriminant to determine the number of equations with real roots.

Case 1: p = 1
For p = 1, the quadratic equation becomes x^2 + qx + 1 = 0.

Substituting the values of q and analyzing the discriminant:
- For q = 1, Δ = 1 - 4(1)(1) = -3 (No real roots)
- For q = 2, Δ = 4 - 4(1)(1) = 0 (One real root)
- For q = 3, Δ = 9 - 4(1)(1) = 5 (Two real roots)
- For q = 4, Δ = 16 - 4(1)(1) = 12 (Two real roots)

Thus, when p = 1, there are two values of q (3 and 4) that result in equations with real roots.

Case 2: p = 2
For p = 2, the quadratic equation becomes 2x^2 + qx + 1 = 0.

Substituting the values of q and analyzing the discriminant:
- For q = 1, Δ = 1 - 4(2)(1) = -7 (No real roots)
- For q = 2, Δ = 4 - 4(2)(1) = -4 (No real roots)
- For q = 3, Δ = 9 - 4(2)(1) = 1 (One real root)
- For q = 4, Δ = 16 - 4(2)(1) = 8 (Two real roots)

Thus, when p = 2, there are two values of q (3 and 4) that result in equations with real roots.

Case 3: p = 3
For p = 3, the quadratic equation becomes 3x^2 + qx + 1 = 0.

Substituting the values of q and analyzing the discriminant:
- For q = 1, Δ = 1 - 4(3)(1) = -11 (No real roots)
- For q = 2, Δ = 4 - 4(3)(1) = -8 (No real roots)
- For q = 3, Δ = 9 - 4(3)(1) = -3 (