Quadratic Equations and Linear Inequalities, Objectives, Roots of a Quadratic Equation, Solving Quadratic Equation by Factorization, Alternative Method

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Recall that an algebraic equation of the second degree is written in general form as . It is called a quadratic equation in .

The coefficient is the first or leading coefficient, ’ is the second or middle coefficient and is the constant term (or third coefficient).

For example,

are all quadratic equations.

Objectives

After studying this lesson, you will be able to:

  • Solve a quadratic equation with real coefficients by factorization and by using quadratic formula;

  • Find relationship between roots and coefficients;

  • Form a quadratic equation when roots are given;

  • Differentiate between a linear equation and a linear inequality;

  • State that a planl region represents the solution of a linear inequality;

  • Represent graphically a linear inequality in two variables;

  • Show the solution of an inequality by shading the appropriate region;

  • Solve graphically a system of two or three linear inequalities in two variables;

Roots of a Quadratic Equation

The value which when substituted for the variable in an equation, satisfies it, is called a root (or solution) of the equation.

If be one of the roots of the quadratic equation

Then

In other words, is a factor of the quadratic equation (1)

Example:

Consider a quadratic equation

Solution:

If we substitute in ,

We get, L.H.S

L.H.S.R.H.S.

Again put in ,

We get, L.H.S

L.H.S.R.H.S.

Again put in ,

We get, L.H.SR.H.S.

and are the only values of x which satisfy the quadratic equation

There are no other values which satisfy

and are the only two roots of the quadratic equation .

Solving Quadratic Equation by Factorization

Recall that you have learnt how to factorize quadratic polynomial of the form , by splitting the middle term and taking the common factors.

Same method can be applied while solving a quadratic equation by factorization.

If and are two factors of the quadratic equation

then

Either or

The roots of the quadratic equation are .

Example:

Using factorization method, solve the quadratic equation:

Solution:

The given quadratic equation is

Splitting the middle term, we have

Or

Or

Either or

Two roots of the given quadratic equation are .

Alternative Method

The given quadratic equation is

This can be rewritten as

Or

Or

The quadratic equation has equal roots .