# Quadratic Equations Notes - Class 10

## Class 10: Quadratic Equations Notes - Class 10

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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
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