Algebra Tutorial-quadratic Formula

Algebra

Tutorial

QUADRATIC

FORMULA

The quadratic formula is used to solve quadratic equations. If you can factor the equation easily you do not use this formula.But there are many quadratic equations which can not be factored Hence the use of Quadratic Formula. There is also another technique called "Completing the Square" You learn that too in this tutorial…. That technique is, in fact, basis for getting or deriving the Quadratic Formula. [The quadratic formula is clumsy and admittedly difficult…but we proceed in simple, small steps.] The Quadratic Formula Quadratic equation: ax.x + b.x + c=0 If a is not zero, it is a quadratic equation---right. You note first the coefficients: a, b and c. The quadratic formula gives the solution: The roots of this equation are: x= ( -b + d) / 2.a

In writing this , What is 'd' " d" is called

and x= (-b -d)/ 2a I have simplified the form here?---d= sqrt(b.b-4ac) discriminant' and has great

Using the

Formula

Let us see

how we can

1 SOLVE:

use the formula

for easy

importance as we will see

in simple steps:

X.X + 5X + 6 =0

Step 1

Identify the three a= 1 b= 5

Step 2

Find the

Subtract: Step 3

of the fromula

Write and

value of d: b.b 4.a.c= 4.1.6= d.d d=1 x= (-b+d)/2a x=(-5+1)/2 x = (-b-d)/2a x= (-5 -1)/2a

The roots are;

x=-2

x=-3

Check:

let us factorise this equation: (x+2)(x+3) =0 x= -2 ; x=-3

coefficients and write them down first: c=6 [Be careful with the of a,b and c.]

25 24 1

x=-2 x= -3

2 I took this simple formula

example to

illustrate the

use of the

in easy steps.

let us see SOLVE

another example: 2X.X +4.X -3 =0 You cannot factorize this easily.

Step 1

Identify the coeffcients a,b,c: a= 2 b= 4 c=-3 Find the value of 'discriminant" 'd': b.b = 16 4.a.c = 4.2.(-3) = -24 <-- note the signs carefully d.d= 16 - (-24) = 40 d= sqrt(40)= 2.sqrt(10)

Step2

Step 3

x= (-b+d)/2a x= (-4+2.sqrt(10))/4 x= -1+ sqrt(10)/2

The other root :

x= -1-sqrt(10)/2

3 SOLVE Step 1 Step 2

d.d=36-28

Step 3

The second root: 4 Here is Solve: Step 1 Step 2 4.a.c= 4.1.4= d.d=0 Note that

Check: Factorise:

x.x -6x+7 =0 Identify the coefficients and write them down: a=1 b= -6 c= 7 find 'd" b.b= 36 4.a.c=4.1.7=28 8 d= 2.sqrt(2) x= (-b+d)/2a x=(6+2sqrt(2))/2 x= 3+sqrt(2) x= 3-sqrt(2) a surprise! x.x+4x+4 =0 Identify the coeffcients: a=1 b=4 Find 'd" b.b= 16 16 d=0 if discriminant 'd' we get two identical x= -b/2a x=-4 / 2 x= -2 x.x+4x+4 =0 (x +2)(x+2) =0 x = -2

c=4

is zero. real roots.

5 Solve Step 1:

x.x - 12x + 36=0 a=1 b=-12

Step 2

Find " d" b.b= 144

4.a.c=4.1.36= d.d=0 Step 3

144 d=0 x= -b / 2a x= 12/2 = 6 (x-6)(x-6) = 0

6 Solve: Step 1 Step 2 4ac=-24 dd = - 8

xx -4x-6=0 a=1 b.b= 16

b=-4

c=36

c=-6

d= imaginary If d.d is negative, We get two complex roots d= 2sqrt(2).i or d= -2sqrt(2)i Step 3

7 Solve Step1 Step 2

x= (-b+d)/2a x= 4 + 2sqrt(2)/2 x= 2+sqrt(2)I x= 2-sqrt(2)i

Another root:

3xx - 9x +3 =0 a=3 b=-9 Find 'd' b.b= 81 36 d=3.sqrt(5) x= (-b+d)/2a x= ( 9+3.sqrt(5))/6 x= 3/2 + sqrt(5)/2 x= 3/2 - sqrt(5)/2

"Completing

the square

4.a.c= 4.3.3 = d.d = 81-36 = 45 Step 3

c= 3

" method

This method is simple to use and is alternative method to using the quadratic formula. In fact, Quadratic formula is derived using this method, as we will see in the next section. 8 Solve: x.x + 3x +2 =0 Step 1 a=1 b=3 c=2 Step 2 x.x + 3x = -2 Take b/2 b/2 =3/2 square this: (9/4) Add this number to both sides of the equation: x.x + 3x + 9/4 = -2 + 9/4 You can see that the left side is a asquare squarebinomila: term: (x+ 3/2) (x+ 3/2) = 1/4

Taking square root Roots are:

x + 3/2 = + 1/2 x= -1 (x + 1) ( x + 2 ) =0

9 Let us see another example: Solve : 2x.x + 16x - 18=0 Divide by 2: x.x + 8x - 9 =0 Step 1 a=1 b=8 Step 2 Take b/2: (b/2)(b/2) Step 3: x.x + 8x + 16 = 9 + 16 (x +4)(x+4) = 25 x+4 = 5 x +4 = -5 Roots are: x=1 x = -9

X + 3/2 = -1/2 x= -2

c=-9 4 16

How we get

the quadratic formula: ax.x + bx + c =0 x.x + (b/a) x = - (c/a) x.x + (b/a) x = - (c/a) For completing the square, add (b.b/4aa) to both sides: (x + b/2a) (x +b/2a) = -(c/a) + b.b/4aa (x+ b/2a) ^2 = (b.b - 4ac)/4aa Taking square root:(x + b/2a) =sqrt(b.b-4ac)/ 2a x= (-b + sqrt(b.b-4ac)) b.b -4ac))/2a x= (-b - sqrt(b.b-4ac)/2a Thus we have obtained or derived' the Quadratic formula. To sum up: If d squared is positive, proceed to get two roots. If d squared is zero, there is only one root : x = -b/2a if d squared is negative, there are two complex roots. Some Solve: 1 2xx +5x -3 =0 2 3xx+ 8x +4=0 3 2xx + 6x + 4=0 4 5xx+7x+2=0 5 xx + 2.5x +1 =0

Exercises

formula.But not be factored

g the Square" the

n simple, small steps.]

learning. later.

hem down first: signs

easily.