Quadratic Equations: Formula, Discriminant and Solutions [Guide]

Quadratic equations are one of the pillars of algebra and represent a central topic in both school and university mathematics. From the quadratic formula to the discriminant, from the connection to the parabola to practical applications in physics and engineering, in this guide we explore every aspect of quadratic equations with numerous worked examples.

What Is a Quadratic Equation

A quadratic equation (or second-degree equation) is an algebraic equation in which the unknown appears with a maximum exponent of 2. The general form, called the standard form or normal form, is:

ax² + bx + c = 0

where:

• a is the coefficient of the second-degree term (a ≠ 0)
• b is the coefficient of the first-degree term
• c is the constant term
• x is the unknown

The condition a ≠ 0 is essential: if a were zero, the equation would reduce to a first-degree equation (bx + c = 0).

Types of Quadratic Equations

Depending on the coefficients, quadratic equations are classified as:

The Quadratic Formula

The quadratic formula allows you to find the solutions (or roots) of any complete quadratic equation:

x = (-b ± √Δ) / (2a)

where Δ (delta) is the discriminant of the equation.

The formula produces two values:

• x&sub1; = (-b + √Δ) / (2a)
• x&sub2; = (-b - √Δ) / (2a)

Derivation of the Formula

The formula is obtained by completing the square. Starting from ax² + bx + c = 0:

1. Divide by a: x² + (b/a)x + c/a = 0
2. Move the constant term: x² + (b/a)x = -c/a
3. Complete the square by adding (b/2a)² to both sides
4. Factor the left side: (x + b/2a)² = (b² - 4ac) / 4a²
5. Take the square root and isolate x

The Discriminant (Δ)

The discriminant is the quantity:

Δ = b² - 4ac

Its value determines the number and nature of the equation's solutions:

Example with Δ > 0: Two Distinct Solutions

Solve: 2x² - 7x + 3 = 0

• a = 2, b = -7, c = 3
• Δ = (-7)² - 4(2)(3) = 49 - 24 = 25 > 0
• x&sub1; = (7 + 5) / 4 = 12/4 = 3
• x&sub2; = (7 - 5) / 4 = 2/4 = 1/2

Example with Δ = 0: Repeated Solution

Solve: x² - 6x + 9 = 0

• a = 1, b = -6, c = 9
• Δ = 36 - 36 = 0
• x = 6 / 2 = 3 (repeated solution)

Note that x² - 6x + 9 = (x - 3)², confirming the repeated solution.

Example with Δ < 0: Complex Solutions

Solve: x² + 2x + 5 = 0

• a = 1, b = 2, c = 5
• Δ = 4 - 20 = -16 < 0
• There are no real solutions
• Complex solutions: x = (-2 ± 4i) / 2 = -1 ± 2i

Solving Pure, Incomplete, and Monomial Equations

Pure Equations (b = 0): ax² + c = 0

Isolate x² and take the square root:

x² = -c/a → x = ±√(-c/a)

Example: 3x² - 27 = 0

• x² = 9
• x = ±3

If -c/a < 0, there are no real solutions (for example, x² + 4 = 0 has no real solutions).

Incomplete Equations with No Constant Term (c = 0): ax² + bx = 0

Factor out x:

x(ax + b) = 0 → x = 0 or x = -b/a

These equations always have two real solutions, one of which is x = 0.

Example: 2x² - 8x = 0

• 2x(x - 4) = 0
• x = 0 or x = 4

Monomial Equations (b = 0, c = 0): ax² = 0

The only solution is x = 0 (repeated solution).

Vieta's Formulas

Vieta's formulas (or relationships between roots and coefficients) establish a direct link between the solutions x&sub1; and x&sub2; and the coefficients of the equation:

• Sum of the roots: x&sub1; + x&sub2; = -b/a
• Product of the roots: x&sub1; · x&sub2; = c/a

These formulas are useful for verifying solutions and for constructing equations from known roots.

Verification Example

For the equation 2x² - 7x + 3 = 0 with solutions x&sub1; = 3 and x&sub2; = 1/2:

• Sum: 3 + 1/2 = 7/2 = -(-7)/2 = 7/2 (verified)
• Product: 3 x 1/2 = 3/2 = 3/2 (verified)

Constructing an Equation from Its Roots

If the roots x&sub1; and x&sub2; are known, the equation is:

x² - (x&sub1; + x&sub2;)x + x&sub1; · x&sub2; = 0

Example: find the equation with roots x = 2 and x = -5:

• Sum = 2 + (-5) = -3
• Product = 2 x (-5) = -10
• Equation: x² + 3x - 10 = 0

Quadratic Equations and the Parabola

The equation y = ax² + bx + c represents a parabola in the Cartesian plane. The solutions of the equation ax² + bx + c = 0 are the x-coordinates of the points where the parabola intersects the x-axis.

Vertex of the Parabola

The vertex has coordinates:

• x_v = -b / (2a)
• y_v = -Δ / (4a)

If a > 0, the parabola opens upward and the vertex is the minimum point. If a < 0, the parabola opens downward and the vertex is the maximum point.

Axis of Symmetry

The parabola is symmetric with respect to the vertical line x = -b/(2a), which passes through the vertex. The two solutions of the equation (when they exist) are symmetric about this axis.

Complex Solutions

When the discriminant is negative, the equation has no solutions in the field of real numbers, but it has two solutions in the field of complex numbers. If Δ < 0:

x = (-b ± i√|Δ|) / (2a)

The two complex solutions are always conjugates of each other, meaning they have the same real part and opposite imaginary parts:

• x&sub1; = α + βi
• x&sub2; = α - βi

where α = -b/(2a) and β = √|Δ|/(2a).

Problems Solvable with Quadratic Equations

Problem 1: Projectile Motion

A ball is thrown vertically with an initial velocity of 20 m/s from a height of 1.5 m. After how many seconds does it hit the ground?

Equation of motion: h(t) = -4.9t² + 20t + 1.5 = 0

• Δ = 400 + 29.4 = 429.4
• t = (-20 ± √429.4) / (-9.8)
• t&sub1; = (-20 + 20.72) / (-9.8) = -0.07 s (not physically meaningful)
• t&sub2; = (-20 - 20.72) / (-9.8) = 4.15 s

Problem 2: Area of a Rectangle

A rectangle has a perimeter of 30 cm and an area of 54 cm². Find the dimensions.

If x and y are the dimensions: 2x + 2y = 30 → y = 15 - x, and x · y = 54 → x(15 - x) = 54

x² - 15x + 54 = 0

• Δ = 225 - 216 = 9
• x = (15 ± 3) / 2
• x = 9 or x = 6
• Dimensions: 9 cm x 6 cm

Problem 3: Financial Investment

A capital of 10,000 EUR invested at compound interest becomes 11,025 EUR in 2 years. What is the annual interest rate?

10,000(1 + r)² = 11,025

(1 + r)² = 1.1025

Let y = 1 + r: y² - 1.1025 = 0 → y = ±1.05

r = 0.05 = 5%

Summary Table: Notable Cases

Frequently Asked Questions (FAQ)

How many solutions does a quadratic equation have?

In the real number field, it can have 0, 1, or 2 solutions, depending on the sign of the discriminant. In the complex number field, it always has exactly 2 solutions (counted with their multiplicity), as guaranteed by the fundamental theorem of algebra.

What happens if the discriminant is zero?

If Δ = 0, the equation has a single real solution, called a repeated solution (or root with multiplicity 2): x = -b/(2a). Geometrically, the parabola is tangent to the x-axis.

How do you verify a solution?

Substitute the found value into the original equation. If the left side equals the right side (i.e., you get 0 = 0), the solution is correct. Alternatively, use Vieta's formulas to verify the sum and product of the roots.

What is the reduced formula?

When b is even (b = 2b'), you can use the reduced formula: x = (-b' ± √(b'² - ac)) / a, with reduced discriminant Δ/4 = b'² - ac. The result is identical but the calculations are simpler.

Does a quadratic equation always have a parabola as its graph?

Yes, the function y = ax² + bx + c (with a ≠ 0) always has a parabola with an axis parallel to the y-axis as its graph. The solutions of the equation ax² + bx + c = 0 correspond to the intersection points of the parabola with the x-axis.

How do you solve a quadratic equation without the formula?

Besides the quadratic formula, you can use: completing the square, factoring (if the coefficients allow it), or the graphical method (identifying the intersections of the parabola with the x-axis). For pure and incomplete equations, there are simpler direct methods.

What are quadratic equations used for in real life?

Quadratic equations model numerous phenomena: projectile motion, free fall of objects, optimization of areas and volumes, calculation of maximum profits in economics, design of arches and bridges in engineering, and many problems in physics and chemistry.