Quadratic Equations - Notes - Class 10

 Page 1


Solving 
Equations 
Page 2


Solving 
Equations 
A quadratic equation is an equation equivalent to one of the form 
 
Where a, b, and c are real numbers and a ? 0  
0
2
? ? ? c bx ax
 
To solve a quadratic equation we get it in the form above 
and see if it will factor. 
6 5
2
? ? x x
Get form above by subtracting 5x and 
adding 6 to both sides to get 0 on right side. 
-5x + 6 -5x + 6 
  
0 6 5
2
? ? ? x x
Factor. 
? ? ? ? 0 2 3 ? ? ? x x
Use the Null Factor law and set each  
factor = 0 and solve. 
0 2 or  0 3 ? ? ? ? x x 3 ? x 2 ? x
  
So if we have an equation in x and the highest power is 2, it is quadratic. 
Page 3


Solving 
Equations 
A quadratic equation is an equation equivalent to one of the form 
 
Where a, b, and c are real numbers and a ? 0  
0
2
? ? ? c bx ax
 
To solve a quadratic equation we get it in the form above 
and see if it will factor. 
6 5
2
? ? x x
Get form above by subtracting 5x and 
adding 6 to both sides to get 0 on right side. 
-5x + 6 -5x + 6 
  
0 6 5
2
? ? ? x x
Factor. 
? ? ? ? 0 2 3 ? ? ? x x
Use the Null Factor law and set each  
factor = 0 and solve. 
0 2 or  0 3 ? ? ? ? x x 3 ? x 2 ? x
  
So if we have an equation in x and the highest power is 2, it is quadratic. 
In this form we could have the case where b = 0. 
0
2
? ? ? c bx ax
Remember standard form for a quadratic equation is: 
0
2
? ?c ax
0 0
2
? ? ? c x ax
When this is the case, we get the x
2
 alone and then square 
root both sides. 
0 6 2
2
? ? x
Get x
2
 alone by adding 6 to both sides and then 
dividing both sides by 2 
+ 6 + 6 
 
6 2
2
? x
  
2 2 
 
3
2
? x
Now take the square root of both 
sides remembering that you must 
consider both the positive and 
negative root. 
 
 
 
 
 
 
? 
 
3 ? ? x
Let's 
check: 
? ? 0 6 3 2
2
? ?
? ? 0 6 3 2
2
? ? ?
0 6 6 ? ? 0 6 6 ? ?
Now take the square root of both 
sides remembering that you must 
consider both the positive and 
negative root. 
 
Page 4


Solving 
Equations 
A quadratic equation is an equation equivalent to one of the form 
 
Where a, b, and c are real numbers and a ? 0  
0
2
? ? ? c bx ax
 
To solve a quadratic equation we get it in the form above 
and see if it will factor. 
6 5
2
? ? x x
Get form above by subtracting 5x and 
adding 6 to both sides to get 0 on right side. 
-5x + 6 -5x + 6 
  
0 6 5
2
? ? ? x x
Factor. 
? ? ? ? 0 2 3 ? ? ? x x
Use the Null Factor law and set each  
factor = 0 and solve. 
0 2 or  0 3 ? ? ? ? x x 3 ? x 2 ? x
  
So if we have an equation in x and the highest power is 2, it is quadratic. 
In this form we could have the case where b = 0. 
0
2
? ? ? c bx ax
Remember standard form for a quadratic equation is: 
0
2
? ?c ax
0 0
2
? ? ? c x ax
When this is the case, we get the x
2
 alone and then square 
root both sides. 
0 6 2
2
? ? x
Get x
2
 alone by adding 6 to both sides and then 
dividing both sides by 2 
+ 6 + 6 
 
6 2
2
? x
  
2 2 
 
3
2
? x
Now take the square root of both 
sides remembering that you must 
consider both the positive and 
negative root. 
 
 
 
 
 
 
? 
 
3 ? ? x
Let's 
check: 
? ? 0 6 3 2
2
? ?
? ? 0 6 3 2
2
? ? ?
0 6 6 ? ? 0 6 6 ? ?
Now take the square root of both 
sides remembering that you must 
consider both the positive and 
negative root. 
 
0
2
? ? ? c bx ax
What if in standard form, c = 0? 
0 0
2
? ? ?bx ax
We could factor by pulling 
an x out of each term. 
0 3 2
2
? ? x x Factor out the common x 
? ? 0 3 2 ? ? x x
Use the Null Factor law and set each  
factor = 0 and solve. 
0 3 2 or  0 ? ? ? x x
2
3
or  0 ? ? x x
If you put either of these values in for x 
in the original equation you can see it 
makes a true statement. 
  
Page 5


Solving 
Equations 
A quadratic equation is an equation equivalent to one of the form 
 
Where a, b, and c are real numbers and a ? 0  
0
2
? ? ? c bx ax
 
To solve a quadratic equation we get it in the form above 
and see if it will factor. 
6 5
2
? ? x x
Get form above by subtracting 5x and 
adding 6 to both sides to get 0 on right side. 
-5x + 6 -5x + 6 
  
0 6 5
2
? ? ? x x
Factor. 
? ? ? ? 0 2 3 ? ? ? x x
Use the Null Factor law and set each  
factor = 0 and solve. 
0 2 or  0 3 ? ? ? ? x x 3 ? x 2 ? x
  
So if we have an equation in x and the highest power is 2, it is quadratic. 
In this form we could have the case where b = 0. 
0
2
? ? ? c bx ax
Remember standard form for a quadratic equation is: 
0
2
? ?c ax
0 0
2
? ? ? c x ax
When this is the case, we get the x
2
 alone and then square 
root both sides. 
0 6 2
2
? ? x
Get x
2
 alone by adding 6 to both sides and then 
dividing both sides by 2 
+ 6 + 6 
 
6 2
2
? x
  
2 2 
 
3
2
? x
Now take the square root of both 
sides remembering that you must 
consider both the positive and 
negative root. 
 
 
 
 
 
 
? 
 
3 ? ? x
Let's 
check: 
? ? 0 6 3 2
2
? ?
? ? 0 6 3 2
2
? ? ?
0 6 6 ? ? 0 6 6 ? ?
Now take the square root of both 
sides remembering that you must 
consider both the positive and 
negative root. 
 
0
2
? ? ? c bx ax
What if in standard form, c = 0? 
0 0
2
? ? ?bx ax
We could factor by pulling 
an x out of each term. 
0 3 2
2
? ? x x Factor out the common x 
? ? 0 3 2 ? ? x x
Use the Null Factor law and set each  
factor = 0 and solve. 
0 3 2 or  0 ? ? ? x x
2
3
or  0 ? ? x x
If you put either of these values in for x 
in the original equation you can see it 
makes a true statement. 
  
0
2
? ? ? c bx ax
What are we going to do if we have non-zero values for 
a, b and c but can't factor the left hand side? 
0 3 6
2
? ? ? x x
This will not factor so we will complete the 
square and apply the square root method. 
First get the constant term on the other side by 
subtracting 3 from both sides. 
3 6
2
? ? ? x x
___ 3 ___ 6
2
? ? ? ? ? x x
We are now going to add a number to the left side so it will factor 
into a perfect square.  This means that it will factor into two 
identical factors.  If we add a number to one side of the equation, 
we need to add it to the other to keep the equation true. 
Let's add 9.  Right now we'll see that it works and then we'll look at how 
to find it. 
9 9 
6 9 6
2
? ? ? x x