Introduction to Quadratic Equations

A quadratic equation is a polynomial equation of degree 2 in one variable.

• It can have two real roots, one repeated real root, or two complex roots.
• Its graph is a parabola, which opens upward if a>0 and downward if a<0.

5x² + x + 2 = 0, x² = 2x + 3 and 2x² = 0 are all examples of quadratic equations.

The following is a real-life example:

Standard Form

The standard form of a quadratic equation is:

ax² + bx + c = 0

Where a, b, and c are constants, x is the variable, and a ≠ 0.

The values of x that satisfy the equation are called the roots (or solutions) of the quadratic equation

Example: For the equation: 2x² + 5x − 3 = 0

• Coefficient of x²: a = 2
• Coefficient of x: b = 5
• Constant term: c = −3

Discriminant

Before solving, you can predict what kind of roots an equation has just by looking at b² − 4ac, called the discriminant:

Quadratic Formula

A method used to find the roots of a quadratic equation, especially when the equation cannot be easily factored. For a quadratic equation of the form ax² + bx + c = 0, the roots are given by:

The ± sign gives the two possible roots of the equation. This formula is also known as the Sridharacharya formula.

Example: Find the roots of the quadratic equation x² − 3x − 4 = 0 using the quadratic formula.

Solution:

Here, a = 1 ,b = −3 and c = −4

Using the quadratic formula: x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substitute the values:

x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)}

x = \frac{3 \pm \sqrt{9 + 16}}{2}

x = \frac{3 \pm \sqrt{25}}{2}

= (3 + 5) / 2 or (3 − 5) / 2

= 8/2 or −2/2

x = 4 or x = −1

Therefore, the roots of the equation are 4 and −1.

Methods to Solve Quadratic Equations

Quadratic equations can be solved using several other methods:

• Factorization: Expresses the equation as a product of two linear factors, each set to zero to find the roots. Quick, but only works when the equation factors easily.
• Completing the square: Rearrange the equation into the form (x−h)² = k to find the roots. Used to derive the quadratic formula and to find a parabola's vertex.

➢Practice: Solved Examples