Class 9 – Quadratic Equations
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Introduction: Greek mathematician Euclid developed a geometrical approach for finding out lengths which, in our present day terminology, are solutions of quadratic equations. Solving of quadratic equations, in general form, is often credited to ancient Indian mathematicians. In fact, Brahmagupta (A.D.598–665) gave an explicit formula to solve a quadratic equation of the form .Sridharacharya (A.D. 1025) derived a formula, known as the quadratic formula,(as quoted by Bhaskara II) for solving a quadratic equation by the method of completing the square. An Arab mathematician Al-Khwarizmi (about A.D. 800) also studied quadratic equations of different types. Abraham bar Hiyya Ha-Nasi, in his book ‘Liber embadorum’ published in Europe in A.D. 1145 gave complete solutions of different quadratic equations.
Quadratic equation: Quadratic equation is a second degree polynomial, which is equal to zero. The general form is where a ≠ 0 and a, b, c R. (For a = 0, the equation becomes a linear equation).
The letters a, b, and c are called coefficients of Quadratic equations
Since the highest power of a quadratic equation is 2, a quadratic equation can have at the most two unique solutions (which are also called the roots of the equation).
Methods of solving Quadratic Equations: There are three Methods of Solving a Quadratic equation.
1. Factorisation method
2. Completing the square root method
3. Quadratic formula
1. Factorization Method:
The following procedure describes the method.
Step1: Let the quadratic equation be .
Step2: Express the quadratic polynomial as the product of two linear
factors, say (px+ q), (rx + s), where p, q, r, s
(This can be done by writing b (coefficient of x)as sum of two quantities whose product is equal to ac )
Now for finding the values of x, we need to use, the zero-product rule: If a.b =0, then
a = 0 or b = 0
Step3: The given equation now becomes (px +q) (rx + s) = 0
The roots of the given quadratic equation are ,
Example: Find the roots of by factorisation method
2. Completing the square method: Let us consideran equation .
If we want to factorize the left hand side of the equation using the method of splitting the middle term, we must determine two integer factors of 4 whose sum is 8. But,the factors of 4 are 1, 4; -1, -4; -2, -2 and 2, 2. In these cases, the sum is not 8. Therefore, using factorization, we cannot solve the given equation . Here we shall discuss a method to solve such quadratic equations.
Solution: We have,
Add and subtracton bothsides
3. Quadratic formula:
Derivinga formula to find the solution of a quadratic equation:
Consider a quadratic equation
To solve this equation we have to convert the terms containing x into a perfect square, i.e. we have to convert into a perfect square as these are the terms containing x.
To make it a perfect square add on both sides
The quadratic equation has got two roots given by
Let and be the 2 unique solutions i.e. roots of the quadratic equation. Thus we have,
Example:Solve using quadratic formula
Solution:On Comparison with the standard quadratic equation we find that
Theroots are 3,.
Sum of the roots: Let are roots of a quadratic equation , then
i.e.,Sum of roots
Product of the Roots: Let are roots of a quadratic equation , then
i.e.,Product of roots
Finding the quadratic equation whose roots are given:
Let are roots of a quadratic equation such that
If are roots of a quadratic equation, then the quadratic equation is.
Example: Find the quadratic equation whose rootsare –3 and 5.
1. Solve by factorization method.
2. Solve by factorization method.
4. Find the sum and product of roots of .
5. Find the quadratic eqution whose roots are 3 and .
Let a and b are the two roots of the quadratic equation . Thus we have,
is a perfect square.