Quadratic Equation

 Quadratic Equation       "practice sums"👈

Definition

A quadratic equation is a polynomial equation of degree 2, which can be written in the standard form:

$$ax^2 + bx + c = 0 \quad \text{where } a \ne 0$$

Here:

• $a, b, c$ are real numbers

• $x$ is the variable

• $a$ is the coefficient of $x^2$
  

---

🔹 Methods to Solve Quadratic Equations

1. Factorisation Method

• Split the middle term into two parts that multiply to give $a \cdot c$ and add to give $b$
  

• Solve by setting each factor to zero

Example:

$$x^2 + 5x + 6 = 0 \Rightarrow (x + 2)(x + 3) = 0 \Rightarrow x = -2, -3$$

---

2. Quadratic Formula

If the equation is not easily factorisable:

$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$

Use when:

• Factorisation is not simple

• Discriminant method is required

---

🔹 Discriminant (D)

$$D = b^2 - 4ac$$

It helps determine the nature of roots:

Discriminant (D)	Nature of Roots
$D > 0$	Real and unequal
$D = 0$	Real and equal
$D < 0$	Imaginary/Complex roots

---

🔹 Nature of Roots Shortcut

If D is:

• Positive & a perfect square → Rational roots

• Positive & not a perfect square → Irrational roots

• Negative → No real solution

---

✅ Example Using Quadratic Formula

Solve:

$$2x^2 + 3x - 5 = 0$$

Here: $a = 2, b = 3, c = -5$

$$D = b^2 - 4ac = 9 + 40 = 49$$
$$x = \frac{-3 \pm \sqrt{49}}{2 \cdot 2} = \frac{-3 \pm 7}{4} \Rightarrow x = 1, -2.5$$