Zeroes of Quadratic Polynomial

The zeroes of a quadratic polynomial p(x) = ax^2 + bx + c are the values of x that satisfy the equation p(x) = 0, i.e., the solutions of ax^2 + bx + c = 0. These are also called the roots of the corresponding quadratic equation.

Formal Definition: A real number k is called a zero of the polynomial p(x) if p(k) = 0. For a quadratic polynomial p(x) = ax^2 + bx + c, the value k is a zero if and only if ak^2 + bk + c = 0.

Geometrical Meaning: The graph of a quadratic polynomial y = ax^2 + bx + c is a parabola. If a > 0, the parabola opens upwards (U-shaped). If a < 0, the parabola opens downwards (inverted U-shaped). The zeroes of the polynomial correspond to the x-coordinates of the points where the parabola crosses or touches the x-axis.

Three Possible Cases:

(i) Two distinct zeroes: The parabola intersects the x-axis at two distinct points. This happens when the discriminant D = b^2 - 4ac > 0. The two zeroes are different real numbers.

(ii) One repeated zero (two equal zeroes): The parabola touches the x-axis at exactly one point (the vertex lies on the x-axis). This happens when D = b^2 - 4ac = 0. The polynomial has two equal zeroes, sometimes called a repeated root or a double root.

(iii) No real zeroes: The parabola does not intersect or touch the x-axis at all. It lies entirely above the x-axis (when a > 0) or entirely below (when a < 0). This happens when D = b^2 - 4ac < 0.

The Discriminant: The expression D = b^2 - 4ac is called the discriminant of the quadratic polynomial. It determines the nature and number of zeroes:
D > 0: Two distinct real zeroes
D = 0: Two equal real zeroes (one repeated zero)
D < 0: No real zeroes (zeroes are complex numbers, beyond the Class 10 scope)

Maximum Number of Zeroes: A polynomial of degree n has at most n zeroes. Since a quadratic polynomial has degree 2, it can have at most 2 zeroes. It may have exactly 2, exactly 1 (repeated), or 0 real zeroes.

Let us derive the quadratic formula by applying the method of completing the square to the general quadratic equation ax^2 + bx + c = 0.

Step 1: Start with the general equation
ax^2 + bx + c = 0, where a is not equal to 0.

Step 2: Divide both sides by a
x^2 + (b/a)x + c/a = 0

This is valid because a is not zero.

Step 3: Move the constant term to the right side
x^2 + (b/a)x = -c/a

Step 4: Complete the square on the left side
To complete the square, add the square of half the coefficient of x to both sides.
Half of b/a is b/(2a). Its square is b^2/(4a^2).

x^2 + (b/a)x + b^2/(4a^2) = -c/a + b^2/(4a^2)

Step 5: Write the left side as a perfect square
(x + b/(2a))^2 = b^2/(4a^2) - c/a

Taking LCM on the right side:
(x + b/(2a))^2 = (b^2 - 4ac) / (4a^2)

Step 6: Take the square root of both sides
x + b/(2a) = +/- sqrt(b^2 - 4ac) / (2a)

Note: sqrt(4a^2) = 2|a| = 2a if a > 0, or -2a if a < 0. In either case, the +/- sign absorbs this, so we write 2a in the denominator.

Step 7: Solve for x
x = -b/(2a) +/- sqrt(b^2 - 4ac) / (2a)

x = (-b +/- sqrt(b^2 - 4ac)) / (2a)

This is the quadratic formula, also known as the Sridharacharya formula, named after the 9th-century Indian mathematician Sridharacharya who first described this method.

Verification with a specific polynomial:
Let p(x) = x^2 - 5x + 6. Here a = 1, b = -5, c = 6.
D = (-5)^2 - 4(1)(6) = 25 - 24 = 1
x = (5 +/- sqrt(1)) / 2 = (5 +/- 1) / 2
x1 = (5 + 1)/2 = 3
x2 = (5 - 1)/2 = 2

Verification: p(3) = 9 - 15 + 6 = 0 and p(2) = 4 - 10 + 6 = 0. Both values are indeed zeroes.

Why the Discriminant Matters:
From the formula x = (-b +/- sqrt(D)) / (2a), we see that the square root of D must be real for real zeroes. If D > 0, sqrt(D) is a positive real number, giving two distinct values. If D = 0, sqrt(D) = 0, giving both values equal to -b/(2a). If D < 0, sqrt(D) is not a real number, so no real zeroes exist.

Zeroes of quadratic polynomials can be classified based on the nature of the discriminant and the methods used to find them.

1. Two Distinct Real Zeroes (D > 0):
When b^2 - 4ac > 0, the polynomial has two different real zeroes. The parabola cuts the x-axis at two distinct points. Example: p(x) = x^2 - 7x + 12 has D = 49 - 48 = 1 > 0, zeroes are x = 3 and x = 4.

2. Two Equal Real Zeroes (D = 0):
When b^2 - 4ac = 0, both zeroes are equal, each being -b/(2a). The parabola just touches the x-axis at its vertex. Example: p(x) = x^2 - 6x + 9 has D = 36 - 36 = 0, both zeroes equal x = 3. Notice that x^2 - 6x + 9 = (x - 3)^2.

3. No Real Zeroes (D < 0):
When b^2 - 4ac < 0, the polynomial has no real zeroes. The parabola lies entirely above or below the x-axis. Example: p(x) = x^2 + x + 1 has D = 1 - 4 = -3 < 0, so it has no real zeroes.

4. Rational Zeroes:
When D is a perfect square, the zeroes are rational numbers. These polynomials can always be factored using the splitting the middle term method. Example: p(x) = 2x^2 + 7x + 3, where D = 49 - 24 = 25. Since sqrt(25) = 5, zeroes are x = (-7 + 5)/4 = -1/2 and x = (-7 - 5)/4 = -3.

5. Irrational Zeroes:
When D > 0 but is not a perfect square, the zeroes are irrational and always occur in conjugate pairs (a + sqrt(b) and a - sqrt(b)). Example: p(x) = x^2 - 2x - 1, where D = 4 + 4 = 8. Zeroes are x = (2 +/- sqrt(8))/2 = 1 +/- sqrt(2).

Methods to Find Zeroes:

(a) Splitting the Middle Term: Best when D is a perfect square and the polynomial can be factored easily with integer or simple rational coefficients.
(b) Quadratic Formula: Works for every quadratic polynomial, regardless of whether it can be factored easily. This is the universal method.
(c) Graphical Method: Plot the polynomial and read off the x-intercepts. This gives approximate values and is useful for visualization.

Zeroes of quadratic polynomials are foundational to many areas of mathematics, science, and engineering.

Physics - Projectile Motion: When an object is thrown upward, its height h at time t is given by a quadratic equation h = -gt^2/2 + ut + h0. The zeroes of this equation give the times when the object is at ground level, helping determine the total time of flight and the landing point.

Economics - Break-Even Analysis: A company's profit P as a function of units sold x often forms a quadratic equation. The zeroes of this polynomial represent the break-even points where the company transitions from loss to profit or vice versa.

Engineering - Structural Design: Parabolic arches and suspension cables follow quadratic equations. Finding the zeroes determines where the arch meets the ground, which is critical for calculating span length and load distribution.

Geometry - Area Problems: When calculating dimensions of shapes with given area constraints, the resulting equations are often quadratic. The zeroes provide the possible dimensions, with negative values being rejected on physical grounds.

Computer Science - Optimization: Quadratic functions appear in optimization algorithms. Their zeroes and vertex help determine minimum or maximum values, which are used in machine learning loss functions and signal processing.

Daily Life - Financial Planning: Compound interest and loan amortization calculations sometimes involve quadratic equations. The zeroes help find the time period needed to achieve a financial goal or pay off a debt.