Quadratic Equations Activity

An equation with one variable, in which  the highest power of the variable is two is called a quadratic equation.

For example, ( i ) 3x2 + 5 x - 8 = 0

(ii) 2y2 - 48 = 0

(iii) 6x2 + 5x = 0

(iv ) y2 = 4 are all quadratic equations. Let us do some activity or problems using quadratic equations.

Activities of Quadratic Equations:

Activity 1 :

Solve the quadratic equation 2x2 - 7x = 39

2x2 - 7x - 39 - 0

`=>`              2x2  - 13x + 6x - 39 = 0  ( factorising the left hand side )

`=>`              x ( 2x - 13 ) + 3 (2 x - 13 ) = 0

`=>`             ( 2x - 13 ) ( x + 3 ) = 0

2x - 13 = 0 or x + 3 = 0

`rArr`              x  =  `13/2`   or  x  =  -3

Hence the quadratic equation is solved by factorisation method.

Activity 2 :

Find the quadratic equation whose solution set is { -2, 3 }

Since solution set is { -2, 3}

we have x = -2 or x = 3

x + 2 = 0 or x - 3 = 0

`rArr`           ( x + 2 ) ( x - 3 ) = 0

`rArr`           x2 - 3x + 2x - 6 = 0

`rArr`           x2 - x - 6 = 0 is the required quadratic equation.

Activity of Quadratic Equations(continued):

Activity 3 :

Solve the quadratic equation 5x2 - 2x - 3 = 0 using the formula.

The roots of the standard quadratic equation ax2 + bx + c = 0 where a`!=` 0, are given by the formula

x   =   `( -b stackrel(+)(-) sqrt ( b^2 - 4ac )) / ( 2a)`

Comparing 5x2 - 2x - 3 = 0 with ax2 + bx + c = 0 we get a = 5, b = -2 and c = -3.

so, x = `(2 stackrel(+)(-) sqrt((-2)^2 - 4. 5. (-3))/(2.5))`

=  `(2 stackrel( +)(-) sqrt ( 64)) / ( 10)`

= `(2 stackrel(+)(-) 8)/10`

= `(2-8)/10` = 1 and `-3/5`

Hence 1 and `(-3)/5` are the roots of the given quadratic equation.

Activity 4 :

Solve the equation 2x4 - 5x2 + 3 + 0 which is reducible to quadratic equation.

Let x2 = y

Then, 2x4 - 5x2 + 3 = 0   `rArr`   2y2 - 5y + 3 + 0

`rArr`   ( y - 1 ) ( 2y - 3 ) = 0

`rArr`   y = 1 or   y   = `3/2`

When y = 1, x2 = 1 `rArr` x = 1 or -1

When y =  `3/2`   x2 = `3/2` `rArr` x = `sqrt(3/2)`   or `-sqrt(3/2)`

Hence the fourth degree equation is solved using the quadratic equation technique.