QUADRATIC EQUATIONS (LECTURE 7)

QUADRATIC EQ.. lesson 7

19TH MAY         LESSON 7

GOOD MORNING  STUDENTS 

YESTERDAY'S LEARNING OUTCOMES 
Recall  solving  word problems based of age and marks 

Today's learning outcomes 
I will be able to solve  more  word  problems .  



lets recall  

Nature of roots of Quadratic equation
$b^2-4ac$ is called the discriminant of the quadratic equation
Fill in the blanks 

S.no	Condition	Nature of roots
1	$b^2-4ac>0$	__________distinct real roots
2	$b^2-4ac=0$	___________ real root
3	$b^2-4ac<0$,	____________ real roots

Question:-

Is the following statement 'True' or 'False'?Justify your answer.

If in a quadratic equation the coefficient of x is zero, then the quadratic equation has no real roots  ?




 Lets try Q6 of Ex 4.3 

Question 6:

The diagonal of a rectangular field is 60 metres more than the shorter side. If the longer side is 30 metres more than the shorter side, find the sides of the field.

LETS SEE THE VIDEO

 Now you understood .Write down the solution 

ANSWER:

Let the shorter side of the rectangle be x m.

Then, larger side of the rectangle = (x + 30) m

However, side cannot be negative. Therefore, the length of the shorter side will be

90 m.

Hence, length of the larger side will be (90 + 30) m = 120 m

Question 7:

The difference of squares of two numbers is 180. The square of the smaller number is 8 times the larger number. Find the two numbers.

ANSWER:

Let the larger and smaller number be x and y respectively.

According to the given question,

However, the larger number cannot be negative as 8 times of the larger number will be negative and hence, the square of the smaller number will be negative which is not possible.

Therefore, the larger number will be 18 only.

Therefore, the numbers are 18 and 12 or 18 and −12.

Tips for quadratic word problems

1. Carefully read down the full problem and understand the situation. Note down what all need to be find out . Choose one of them as the variable x
2. Now looks for the condition, operations given in the word problem and find all the other quantities in term of variable x
3. Here are some of the operation which you can find in word problem and their mathematical meaning
   Addition: added to, combined, increased by, more than, sum, total, from now
   Subtraction: decreased by, difference of, less than,ago
   Multiplication: increased by a factor of, multiplied by, times,product
   Division: out of, per, ratio of
   Equals: are, gives, is, will be
4. Now go for the condition given in the problem and formula the mathematical equation
5. You can rearrange them to form of quadratic equation
6. Now you can solve the equation using factorization  or quadratic formula
7. Both the roots may not satisfy the word problem, so always verify it with the problem.Many times negative roots are not the solution of word problem

Question 1
A rectangle has a length that is 2 less than 3 times the width. If the area of this rectangle is 16, find the dimensions and the perimeter.

Question 2
X and Y working together can complete a certain project is 6 days. If X alone works on the project, He will take 5 days less than Y to complete. Find days required for X and Y to complete the project alone?


ANSWERS
Fill in the blanks:-   Two,One ,No 

Solution
False, since the discriminant in this case is -4ac which can still be non negative if a and c are of opposite signs or if one of a or c is zero


 solution 1 1Let x= the length of the rectangle and y= its width
As per the question
$x=3y-2$
Given ,The area of this rectangle is 16
Now
$Area=xy=16$
or
$(3y-2)y=16$
$3y^2-2y-16=0$

Factoring
$(3y-8)(y+2)=0$
y = 8/3 and -2
Since we can't have a negative width, y= 8/3
Now x=3y-2 = 6
Perimeter = 2x + 2y = 12 + 16/3 = 64/3

solution 2 Let p and q are days required for X and Y to complete the project alone
Now p =q-5
Also
$\frac{1}{p}+\frac{1}{q}=\frac{1}{6}$
or
$\frac{1}{q-5}+\frac{1}{q}=\frac{1}{6}$
$\frac{q+q-5}{q(q-5)}=\frac{1}{6}$
$6(2q-5)=q^2-5q$
$q^2-17q+30=0$
factoring this quadratics
q=15 or 2
It can not be 2 as then p will be negative
So q=15 days and p=10 days

THANK YOU AND HAVE A NICE DAY 

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