Quadratic Equation Basics
A second-degree polynomial equation that can be written in the standard form ax2 + bx + c = 0 where x is the single variable, and a, b, and c are constants (numbers), with the essential condition that a ≠ 0, is called a quadratic equation in one variable.
Since it is an algebraic equation of degree 2, meaning the highest power of the variable is 2, it can have at most two solutions, called roots, depending on the values of the coefficients.
Key Characteristics :
🔸 Standard Form : ax2 + bx + c = 0.
🔸 Degree of 2 : The highest power of the variable is 2 (the x2 term).
Coefficients :
🔸 a : The leading coefficient (cannot be zero).
🔸 b : The coefficient of the x term (can be zero).
🔸 c : The constant term (can be zero).
Solution of a Quadratic Equation :
If a real number α is called a root of the quadratic equation ax2 + bx + c = 0, a ≠ 0, then it must satisfy the condition aα2 + bα + c = 0.
We also say that x=α is a solution of the quadratic equation, or that α satisfies the quadratic equation.
Note: The zeroes of the quadratic polynomial ax2 + bx + c and the roots of the quadratic equation ax2 + bx + c = 0 are the same.
👉 Check whether the following 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
👉 Represent the following situations in the form of quadratic equations :
🔹 1. The area of a rectangular plot is 528 m2. The length of the plot is one more than twice its breadth. We need to find the length and breadth of the plot.
🔹 2. The product of two consecutive positive integers is 306. We need to find the integers.
🔹 3. Rohan’s mother is 26 years older than him. The product of their ages 3 years from now will be 360. We would like to find Rohan’s present age.
🔹 4. A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
👉 Let us solve
🔹 1.
🔹 2.
🔹 3. , a ≠ 0
🔹 4. , a ≠ 0
🔹 5. , a ≠ 3, -5