Quadratic Expressions and Equations

2.1 Quadratic Expressions.

a.Identifying quadratic expressions.

A quadratic expression is an algebraic expression of the form ax base 2 + bx + c, where a,b,c are constants.The highest power of the unknown x is 2.For example:

3x base 2 + 5x -1 is a quadratic expression.

Notes.

-The highest power of the unknow X must be above 2.

-The unknows must be same, example: 5y base 2 -2y+7.

b.Forming quadratic expresssions by multiplying any two linear expressions.

When two any two linear expressions with the same unknown are multiplied, a quadratic expressions is formed.

Example:

(x+2)(2x-3)

= X x 2X X x 3 + 2 x 2X 2 x 3

=2X base 2 3X + 4X 6

=2X base 2 + X -6

(n-3)base 2 = (n-3)(n-3)

= n base2 3n- 3n + 9

= n base2 6n + 9

7p base2 (p+1) base 2

=7p base 2 (p+1)(p+1)

=7p base2 (pbase2 + p + p + p + 1)

=7p base2 (pbase2 + 2p + 1)

= 7p base2 pbase2 2p -1

=6pbase2 2p -1

c.Forming quadratic expressions based on specific situations.

To form quadratic expressions based on specific situations, follow the steps below:

1.Choose an alphabet, for example, x, to represent an unknown in the given situation

If the symbol is not given.

2.Express the other unknown in terms of x.

3.Form a quadratic expression based on the situation given.

Example: The breadth of a rectangular sheet of paper is 2cm less than its length.Express the area of the paper as a quadratic expression.

Solution:

Let the length of the paper be x cm. Therefore, its breadth is (x-2)cm.

Area of the paper= x(x-2)

= (xbase2 2x) cm

2.2 Factorisation of quadratic expresions.

a.Factorising quadratic expressions of the forms ax base 2 + bx and ax base2 + c

1.Factorisation of a quadratic expression is the process of writing the quadratic expressions as a product of two linear exprisons

-Example,

Xbase 2- 1 = (x+1)(x-1)

Xbase 2 + 3x + 2 = (X+1)(X+2)

2.Quadratic expressions of the form axbase2+bx or axbase 2+ c can be factorized if the terms in the expressions have common factors except 1.

For example:

a. 2X base 2 + 4

=2(xbase2 + 2)

b. 2x base 2+7 cannot be factorized because there is no common factor for 2x base 2 and 7 execpt 1.

b.Factorising quadratic expressions of the form pxbase2 q, where p and q are perfect squares.

1. A square number sometimes also called a perfect square. For example 1,4,9,16..

2. Quadratic expressions of the form px base 2 q where p and q are perfect squares can be factorized by using the identity a base 2 b base2= (a+b)(a-b), that is, the difference between two squares.

Example:

. 9X base 2 16= (3X)base2 (4)base 2

=(3X + 4)(3X-4)

c.Factorising quadratic expressions of the form aX base2 + bx +c, where a,b and c does not equal to 0.

.Quadratic expression of the form aX base 2 + bx + c can be factorized by the cross method or the inspection method.

d.Factorising quadratic expressions containing coefficients with common factors.

.To factorise quadratic expressions containing coefficients with common factors, first extract the common factors and then factorise the reduced quadratic expressions. By doing so, the factorization will be done completely.

2.3Quadratic Equations

a.Identifying quadratic equations 1 unknown.

-A quadratic equation in one unknown is an equation that contains only one unknown and the highest power of the unknown is 2. For example, 2X base 2 + 5X +2=0

(Highest power of the unknown X is 2)

b.Writing quadratic equations in general form as aX base 2 + bx + c=0

Example: y(y+2)=3

Ybase 2 + 2y = 3

Y base + 2Y 3 = 3-3

Y base 2 + 2y 3 = 0

c.Forming quadratic equations based on specific situations.

To form quadratic equations based on specific situations, follow the steps below:

1.Choose an alphabet, for example, x to represent an unknown in the given situation if the symbol is not given.

2.Express the other unknown in terms of X

3.Form a quadratic equation based on the situation given.

Example: Ravi is 5 years younger than Rosli.The product of their ages is 234 years, Form a quadratic equation in general form based on the above information.

2.4Roots of Quadratic Equations

a.Determining whether a given value is a root of a quadratic equation.

1. A root of a quadratic equation is the value of the unknown which satisfies the quadratic equation.

2. To determine whether a given value is a root of a quadratic equation, substitute the given value for the unknown in the quadratic equation. If both sides of the equation have the same value, then the given value is a root of the quadratic equation. Conversely, if both sides of the equation have different values, then the given value is not a root of the quadratic equation.

b.Determining the solution of quadratic equations by the trial and error method

1.The solution of a quadratic equation is also known as the roots of that equation.

2. One method to determine the solution of quadratic equation is to guees a solution and then substitute it into the equation. This method is known as trial and error method.

3. To find the roots for quadratic equations of the form Xpower2 + bx + c=0 by trial and error method, choose the values which adre factors of c.

c.Determining the solutions of quadratic equations by factorization.

1. We can also determine the solutions of the quadratic equations by factorization.

2. To solve a quadratic equation by the factorization method, follow the steps below:

a.Express the quadratic equation in the general form axpower2 + bx + c=0

b. Factorise the quadratic expression axpower2 + bx + c as the product of two linear expressions, that is (mx + P)(nx + q)

c. Solve the linear equations.

Mx+P=0 or nx+q=0

X= - p over m x= - q over n

d.Solving problem involving quadratic equations.

To slove problems involving quadratic equations, follow the steps below:

1. Form a quadractic equation based in the situation given.

2. Solve the quadratic equation

3.If there are two roots, choose the one which is meaningful or rational.

Common Mistakes

1.Failure to factorise a quadratic equation completely.

Example: factorise 2y