152
edits
Changes
From Karnataka Open Educational Resources
→Hints for difficult problems
Interpretation of the Problem:
Interpretation of the Problem:
#Compare the equation with standard form and identify the values of a,b,c
#Compare the equation with standard form and identify the values of a,b,c
#To find the sum of the roots of the quadratic equation using the formula
#To find the sum formformof the roots of the quadratic equation using the formula
#To find the product of the roots of the equation
#To find the product of the roots of the equation
# Using the identity & rewriting p^3+q^3 as (p+q)^3-3pq(p+q)
# Using the identity & rewriting p^3+q^3 as (p+q)^3-3pq(p+q)
Therefore,<br>
Therefore,<br>
p^3+q^3=(p+q)^3-3pq(p+q)<br> =(2)^3-3(1/2)(2)<br>=
p^3+q^3=(p+q)^3-3pq(p+q)<br> =(2)^3-3(1/2)(2)<br>=
8-3<br>=5
8-3<br>=5,<br>
Following explains the steps and gives examples of solving by completing the square. It also shows how the Quadratic Formula is generated by this process. So I'll just do just one example of the process in this lesson. If you need further feel free to reach me.<br>
PROBLEM 2: Solve x2 – 4x – 8 = 0. By completing the square.<br>
Interpretation of the problem:<br>
Is it a quadratic equation?<br>
Knowledge about coefficients of the variable.<br>
Knowledge of steps for completing the given equation as square.<br>
Knowledge of root.<br>
Different approaches to solve:<br>
Factorization Method (sometimes in few ex as x2+6x-7=0)<br>
Completing the square.<br>
As noted above, this quadratic does not factor, so I can't solve the equation by factoring. And they haven't given me the quadratic in a form that is ready to square-root. But there is a way for me to manipulate the quadratic to put it into that form, and then solve. It works like this:<br>
1) First, I put the loose number “8” on the other side of the equation:<br>
x2 – 4x – 8 = 0 <br>
x2 – 4x = 8<br>
2) Then I look at the coefficient of the x-term, which is –4 in this case. I take half of this number (including the sign), giving me –2. Then I square this value to get +4, and add this squared value to both sides of the equation:<br>
x2 – 4x + 4 = 8 + 4 <br>
x2 – 4x + 4 = 12 <br>
3) This process creates a quadratic that is a perfect square, and factoring gives me:<br>
(x – 2)2 = 12<br>
Tip : I know it's a "minus two" inside the parentheses because half of –4 is –2. If you note the sign when you're finding one-half of the coefficient, then you won't mess up the sign when you're converting to squared-binomial form.<br>
4) Now I can square-root both sides of the equation, simplify, and solve:<br>
(x – 2)2 = 12<br>
Then the solution is<br>
For each approach:<br>
Prior knowledge:<br>
About quadratic equation <br>
: About co-efficient’s <br>
: Comparing the equation with standard form<br>
: Dividing and squaring the value of ‘b’<br>
Gap identification:<br>
Recalling the standard equation.<br>
: Identifying the values of a,b,c in given equation<br>
: Dividing and squaring<br>
Algorithm :<br>
1. Equating the equation to zero or standard form.<br>
2. Translating loose number “ c” i.e. constant on the other side of the equation<br>
3. Making half and squaring the co-efficient of variable (x).<br>
4. Add the square on both sides of equation.<br>
5. Put in the complete square form. (a+b)2 or (a-b) 2<br>
6. Find the value of variable ‘x’ /square-roots of the variable<br>
=Ex.no.9.11 /problem no.9=
=Ex.no.9.11 /problem no.9=
