Quadratic equations play a significant role in solving problems that involve numbers, geometry, motion, and measurement. For instance, quadratic equations can be used to find two unknown numbers when certain conditions about the two numbers are given. Also, quadratic equations can be used to find certain dimensions like length or breadth of a rectangle when certain conditions about its area or perimeter are given.
In many problems, when distance, speed, and time are involved, quadratic equations can be formed to find certain unknown quantities. Quadratic equations can be used to solve problems that involve certain areas of geometric shapes when the dimensions of those shapes are not known.
In many problems, when distance, speed, and time are involved, quadratic equations can be formed to find certain unknown quantities. Quadratic equations can be used to solve problems that involve certain areas of geometric shapes when the dimensions of those shapes are not known.
Working rule to solve application problems are as follows:
Step -1: The problem is first expressed into mathematical statement by forming a quadratic equation.
Step - 2: After obtaining a quadratic equation, it can be easily solved either by using the method of factorisation or by using the quadratic formula to find the unknown quantities.
Factorisation method:
Let us see the procedure to solve the quadratic equation by the factorisation method.
Step 1: Write the given equation in standard form.
Step 2: Express the middle term as the sum of two terms such that the sum satisfies the middle term, and the product should satisfy the extreme product.
Step 3: Group the expression into two linear factors by taking the common term outside.
Step 4: Now, solve for \(x\) by equating each linear factor to zero. The obtained values of \(x\) are called the roots or zeroes of the equation.
Step 2: Express the middle term as the sum of two terms such that the sum satisfies the middle term, and the product should satisfy the extreme product.
Step 3: Group the expression into two linear factors by taking the common term outside.
Step 4: Now, solve for \(x\) by equating each linear factor to zero. The obtained values of \(x\) are called the roots or zeroes of the equation.
Quadratic formula:
The formula for finding the roots of the quadratic equation \(ax^2 + bx + c = 0\) is:
This formula is known as the quadratic formula.