Quadratic Equation in One Variable
🎯 What will you learn in this chapter?
👉 The standard form of a quadratic equation ax² + bx + c = 0,
👉 The main characteristics of a quadratic equation
👉 The basic idea of solving a quadratic equation
👉 Some problems on quadratic equations and their solutions
The standard form of a quadratic equation is ax2 + bx + c = 0, where x is an a quantity whose exact value is not fixed or known. or variable, and a (≠ 0), b and c are constants. This is called a degree 2 equation in one variable.
📚 In x2, x is called the base and 2 is called the exponent or power.
📚 In ax2 + bx + c = 0, a and b are respectively called the coefficients of x2 and x.
Key Characteristics:
🔸 Standard form: ax2 + bx + c = 0
🔸 Highest power of the variable: 2 that is, the power of x2.
Coefficients:
🔸 a: leading coefficient or coefficient of x2, where a ≠ 0.
🔸 b: coefficient of the x term.
🔸 c: constant term.
📚 Easy Rule to Identify a Quadratic Equation
An equation may appear to be quadratic at first, but that is not enough. First, simplify the equation and bring it to its standard form.
After writing it in the form ax² + bx + c = 0, if the coefficient of the x² term is not zero, then the equation is a quadratic equation.
Solution of a Quadratic Equation:
🔸 If a real number α is a root of the quadratic equation ax2 + bx + c = 0, a ≠ 0, then it must satisfy the condition aα2 + bα + c = 0.
🔸 In this case, we also say that x = α is a solution of the quadratic equation, or the value α makes the equation true. the quadratic equation.
🔸 If α is a root of the quadratic equation ax2 + bx + c = 0, a ≠ 0, then (x - α) is a factor of the expression. The converse is also true.
🔸 Since the highest power of the variable in ax2 + bx + c = 0 is 2, it can have at most two solutions depending on the values of the coefficients. These solutions are called roots.
📚 The zeroes of f(x)=ax2 + bx + c and the roots of ax2 + bx + c = 0 are the same.
👇 Check whether the following equations are quadratic equations:
🔹 1. (x – 2)2 + 1 = 2x – 3
🔹 2. x(x + 1) + 8 = (x + 2) (x – 2)
🔹 3. x (2x + 3) = x2 + 1
🔹 4. (x + 2)3 = x3 – 4
🔹 5. (2x – 1)(x – 3) = (x + 5)(x – 1)
🔹 6. x2 + 3x + 1 = (x – 2)2
🔹 7. (x + 2)3 = 2x (x2 – 1)
🔹 8. x3 – 4x2 – x + 1 = (x – 2)3
👇 Express the following situations in the form of quadratic equations:
🔹 1. The area of a rectangular field is 528 m2. Its length is 1 more than twice its breadth. Find the length and breadth of the field.
🔹 2. The product of two consecutive positive integers is 306. Find the integers.
🔹 3. Rohan’s mother is 26 years older than him. After 3 years, the product of their ages will be 360. Find Rohan’s present age.
🔹 4. A train travels a distance of 480 km at a uniform speed. If its speed were 8 km/h less, it would take 3 hours more to cover the same distance. Find the speed of the train.
👇 Let’s solve:
🔹 1.
🔹 2. , x ≠ 0
🔹 3. , x ≠ 3, −5
👉 Click for more solved problems
💡 Common Mistakes
| Wrong Idea | Correct Idea |
|---|---|
| An equation is quadratic whenever it contains x². | After simplification, the x² term must remain. |
| The equation is still quadratic even when a = 0. | If a = 0, the equation is no longer quadratic. |
| Every quadratic equation has two distinct roots. | It may have two distinct roots, two equal roots, or no real roots. |
| Roots and factors are the same thing. | A root is a value of x, while a factor is a multiplicative part of an expression. |
💡 Quick Revision
🔹 Standard form of a quadratic equation: ax² + bx + c = 0
🔹 Here a ≠ 0
🔹 The highest power of the variable is 2
🔹 a is the coefficient of x²
🔹 b is the coefficient of x
🔹 c is the constant term
🔹 A quadratic equation can have at most two roots
💡 Questions and Answers (FAQ)
An equation in one variable whose highest power is 2 and which can be written in the form ax² + bx + c = 0 is called a quadratic equation. Here a ≠ 0.
The standard form of a quadratic equation is ax² + bx + c = 0, where a, b and c are constants and a ≠ 0.
Here a is the coefficient of x², b is the coefficient of x and c is the constant term.
No. If a = 0, the x² term will not remain and the equation will no longer be quadratic. So, in a quadratic equation, a ≠ 0.
A value of the variable that satisfies the quadratic equation is called a root or solution of the quadratic equation.
In general, a quadratic equation can have at most two roots.
Quadratic equations are very important for understanding algebra, factorisation, equation solving, graphs, physics problems and higher mathematics.
Quadratic equations are commonly solved by factorisation, completing the square, using the quadratic formula and sometimes by using graphs.