Quadratic Equation: Solving Quadratic Equations-Factor Method, Quadratic Formula (Part 2) (For CBSE, ICSE, IAS, NET, NRA 2022)

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Solution of a Quadratic Equation

The value of the variable for which LHS and RHS of the quadratic equation become equal is called a root or solution of the quadratic equation. There are two algebraic methods for finding the solution of a quadratic equation-

(i) Factor Method- In this method, the equation is factorized into its linear factors and then the value of the variable is found out.

Factor Method

Example – Solve the equation

Sol- Firstly, it will be factorized by splitting the middle term.

Therefore, and are the solutions of the given quadratic equation.

(ii) Quadratic Formula- Here we use a formula to find the solution of a quadratic equation. The formula for finding of two solutions or roots are-

The sign implies that we have two roots of : the one is computed by putting sign while the other is found out by putting sign. That is, and

Example- Solve the equation using quadratic formula

Sol-

On taking positive sign

On taking negative sign

Therefore, the roots of the equation are and .

In the quadratic formula for the roots of the equation, the expression is denoted by D and is called Discriminant, because it determines the number of solutions or nature of roots of a quadratic equation.

For a quadratic equation if

(i) , the equation has two real distinct roots, which are-

and

(ii) , then equation has two real equal roots, each equal to

(iii) , the equation will not have any real root, since square root of a

negative real number is not a real number.

Thus, a quadratic equation will have at the most two roots.

Example- Without determining the roots, comment on the nature (number of solutions)

of roots of the following equations-

(i)

(ii)

(iii)

Sol- (i)

, Therefore, the equation has two real distinct roots.

(ii)

, Therefore, the equation does not have any real roots.

(iii)

, Therefore, the equation has two real equal roots.

1. The sum of two numbers is . If the sum of their reciprocals is . Find the numbers.

Solution- Let the first number be

Second number will be

The reciprocal of the first number will be

The reciprocal of the second number will be

As per the question,

On cross multiplication, we get,

If we take the first number as , then second number will be . If we take first number as , then second number will be .

Therefore, the required numbers are and .

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