Quadratic Formula | द्विघात समीकरण | Quadratic Equation Tricks | How To Solve Quadratic Equations

 Quadratic Formula | द्विघात समीकरण | Quadratic Equation Tricks | How To Solve Quadratic Equations

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An example of a Quadratic Equation:

The function makes nice curves like this one:

Name

The name Quadratic comes from "quad" meaning square, because the variable gets  squared (like x²).

It is also called an "Equation of Degree 2" (because of the "2" on the x)

Youtube Video: Quadratic Formula

Standard Form

The Standard Form of a Quadratic Equation looks like this:

• a, b and c are known values. a can't be 0.

• "x" is the variable or unknown (we don't know it yet).

 

Here are some examples:

2x2 + 5x + 3 = 0		In this one a=2, b=5 and c=3
x2 − 3x = 0		This one is a little more tricky: Where is a? Well a=1, as we don't usually write "1x2" b = −3 And where is c? Well c=0, so is not shown.
5x − 3 = 0		Oops! This one is not a quadratic equation: it is missing x2 (in other words a=0, which means it can't be quadratic) Youtube Video: Quadratic Formula Hidden Quadratic Equations! As we saw before, the Standard Form of a Quadratic Equation is ax2 + bx + c = 0 But sometimes a quadratic equation doesn't look like that! For example: In disguise In Standard Form a, b and c x2 = 3x − 1 Move all terms to left hand side x2 − 3x + 1 = 0 a=1, b=−3, c=1 2(w2 − 2w) = 5 Expand (undo the brackets), and move 5 to left 2w2 − 4w − 5 = 0 a=2, b=−4, c=−5 z(z−1) = 3 Expand, and move 3 to left z2 − z − 3 = 0 a=1, b=−1, c=−3

About the Quadratic Formula

Plus/Minus

First of all what is that plus/minus thing that looks like ± ?

The ± means there are TWO answers:

x = −b + √(b² − 4ac)2a

x = −b − √(b² − 4ac)2a

Here is an example with two answers:

But it does not always work out like that!

• Imagine if the curve "just touches" the x-axis.
• Or imagine the curve is so high it doesn't even cross the x-axis!

This is where the "Discriminant" helps us ...

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Discriminant

Do you see b² − 4ac in the formula above? It is called the Discriminant, because it can "discriminate" between the possible types of answer:

• when b² − 4ac is positive, we get two Real solutions
• when it is zero we get just ONE real solution (both answers are the same)
• when it is negative we get a pair of Complex solutions

Complex solutions? Let's talk about them after we see how to use the formula.

Youtube Video: Quadratic Formula

 

Using the Quadratic Formula

Just put the values of a, b and c into the Quadratic Formula, and do the calculations.

Example: Solve 5x² + 6x + 1 = 0

Coefficients are:a = 5, b = 6, c = 1

Quadratic Formula:x = −b ± √(b² − 4ac)2a

Put in a, b and c:x = −6 ± √(6² − 4×5×1)2×5

Solve:x = −6 ± √(36 − 20)10

 x = −6 ± √(16)10

 x = −6 ± 410

 x = −0.2 or −1

 

Answer: x = −0.2 or x = −1

 

And we see them on this graph.

Check -0.2:		5×(−0.2)2 + 6×(−0.2) + 1 = 5×(0.04) + 6×(−0.2) + 1 = 0.2 − 1.2 + 1 = 0
Check -1:		5×(−1)2 + 6×(−1) + 1 = 5×(1) + 6×(−1) + 1 = 5 − 6 + 1 = 0

Example: Solve x² − 4x + 6.25 = 0

Coefficients are:a=1, b=−4, c=6.25

Note that the Discriminant is negative:b² − 4ac = (−4)² − 4×1×6.25
              = −9

Use the Quadratic Formula:x = −(−4) ± √(−9)2

√(−9) = 3i
(where i is the imaginary number √−1)

So:x = 4 ± 3i2

 

Answer: x = 2 ± 1.5i

 

The graph does not cross the x-axis. That is why we ended up with complex numbers.

BUT an upside-down mirror image of our equation does cross the x-axis at 2 ± 1.5 (note: missing the i).

Just an interesting fact for you!

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Youtube Video: Quadratic Formula

A quadratic equation is a second-order polynomial equation in a single variable 

(1)

with . Because it is a second-order polynomial equation, the fundamental theorem of algebra guarantees that it has two solutions. These solutions may be both real, or both complex.

Among his many other talents, Major General Stanley in Gilbert and Sullivan's operetta the Pirates of Penzance impresses the pirates with his knowledge of quadratic equations in "The Major General's Song" as follows: "I am the very model of a modern Major-General, I've information vegetable, animal, and mineral, I know the kings of England, and I quote the fights historical, From Marathon to Waterloo, in order categorical; I'm very well acquainted too with matters mathematical, I understand equations, both the simple and quadratical, About binomial theorem I'm teeming with a lot o' news-- With many cheerful facts about the square of the hypotenuse."

The roots  can be found by completing the square,

(2)

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(3)

(4)

Solving for  then gives

(5)

This equation is known as the quadratic formula.

Here are the steps required to solve a quadratic using the quadratic formula:

Youtube Video: Quadratic Formula