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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Youtube Video: Quadratic Formula

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 ...

Youtube Video: Quadratic Formula

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)

src="https://mathworld.wolfram.com/images/equations/QuadraticEquation/NumberedEquation3.gif" style="border: 0px;" width="217" />

(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

Scientific notation │Scientific Notation in hindi │What is Scientific Notation?

Scientific notation │Scientific Notation in hindi │What is Scientific Notation?

 Scientific notation │Scientific Notation in hindi │What is Scientific Notation? 

Scientific notation is a way of writing very large or very small numbers. A number is written in scientific notation when a number between 1 and 10 is multiplied by a power of 10. For example, 650,000,000 can be written in scientific notation as 6.5 ✕ 10^8.

Youtube Video:- Scientific Notation

Scientific Notation (also called Standard Form in Britain) is a special way of writing numbers:

Like this:
Or this:

It makes it easy to use big and small values.

OK, How Does it Work?

Example: 700

Why is 700 written as 7 × 10² in Scientific Notation ?

700 = 7 × 100

and 100 = 10² (see powers of 10)

so 700 = 7 × 10²

Both 700 and 7 × 10² have the same value, just shown in different ways.

Example: 4,900,000,000

    1,000,000,000 = 10⁹ ,
so 4,900,000,000 = 4.9 × 10⁹ in Scientific Notation

The number is written in two parts:

• Just the digits, with the decimal point placed after the first digit, followed by
• × 10 to a power that puts the decimal point where it should be
  (i.e. it shows how many places to move the decimal point).

In this example, 5326.6 is written as 5.3266 × 10³,
because 5326.6 = 5.3266 × 1000 = 5.3266 × 10³

Other Ways of Writing It

3.1 × 10^8

We can use the ^ symbol (above the 6 on a keyboard), as it is easy to type.

Example: 3 × 10^4 is the same as 3 × 10⁴

• 3 × 10^4 = 3 × 10 × 10 × 10 × 10 = 30,000

Calculators often use "E" or "e" like this:

Example: 6E+5 is the same as 6 × 10⁵

• 6E+5 = 6 × 10 × 10 × 10 × 10 × 10 = 600,000

Example: 3.12E4 is the same as 3.12 × 10⁴

• 3.12E4 = 3.12 × 10 × 10 × 10 × 10 = 31,200

How to Do it

To figure out the power of 10, think "how many places do I move the decimal point?"

	When the number is 10 or greater, the decimal point has to move to the left, and the power of 10 is positive.
	When the number is smaller than 1, the decimal point has to move to the right, so the power of 10 is negative.

Example: 0.0055 is written 5.5 × 10^-3

Because 0.0055 = 5.5 × 0.001 = 5.5 × 10^-3

Example: 3.2 is written 3.2 × 10⁰

We didn't have to move the decimal point at all, so the power is 10⁰

But it is now in Scientific Notation

Check!

After putting the number in Scientific Notation, just check that:

• The "digits" part is between 1 and 10 (it can be 1, but never 10)
• The "power" part shows exactly how many places to move the decimal point

Why Use It?

Because it makes it easier when dealing with very big or very small numbers, which are common in Scientific and Engineering work.

Example: it is easier to write (and read) 1.3 × 10^-9 than 0.0000000013

It can also make calculations easier, as in this example:

Example: a tiny space inside a computer chip has been measured to be 0.00000256m wide, 0.00000014m long and 0.000275m high.

What is its volume?

Let's first convert the three lengths into scientific notation:

• width: 0.000 002 56m = 2.56×10^-6
• length: 0.000 000 14m = 1.4×10^-7
• height: 0.000 275m = 2.75×10^-4

Then multiply the digits together (ignoring the ×10s):

2.56 × 1.4 × 2.75 = 9.856

Last, multiply the ×10s:

10^-6 × 10^-7 × 10^-4 = 10^-17 (easier than it looks, just add −6, −4 and −7 together)

The result is 9.856×10^-17 m³

It is used a lot in Science:

Example: Suns, Moons and Planets

The Sun has a Mass of 1.988 × 10³⁰ kg.

Easier than writing 1,988,000,000,000,000,000,000,000,000,000 kg
(and that number gives a false sense of many digits of accuracy.

It can also save space! Here is what happens when you double on each square of a chess board:

Values are rounded off, so 53,6870,912 is shown as just 5×10⁸

That last value, shown as 9×10¹⁸ is actually 9,223,372,036,854,775,808

Engineering Notation

Engineering Notation is like Scientific Notation, except that we only use powers of ten that are multiples of 3 (such as 10³, 10^-3, 10¹² etc).

Examples:

• 2,700 is written 2.7 × 10³
• 27,000 is written 27 × 10³
• 270,000 is written 270 × 10³
• 2,700,000 is written 2.7 × 10⁶

Example: 0.00012 is written 120 × 10^-6

Notice that the "digits" part can now be between 1 and 1,000 (it can be 1, but never 1,000).

The advantage is that we can replace the ×10s with Metric Numbers. So we can use standard words (such as thousand or million), prefixes (such as kilo, mega) or the symbol (k, M, etc)

Example: 19,300 meters is written 19.3 × 10³ m, or 19.3 km

Example: 0.00012 seconds is written 120 × 10^-6 s, or 120 microseconds

वर्ग एवं वर्गमूल के सवाल एवं सूत्र – Square And Square Roots

वर्ग एवं वर्गमूल (Square And Square Roots)-  इस अध्याय के अंतर्गत संख्याओं के वर्ग एवं वर्गमूल से सम्बंधित प्रश्न पूछे जाते है जिनमे वर्ग मूल वाली संख्या का दोहरा वर्ग मूल ज्ञात करना किसी संख्या का वर्गमूल देकर कोई व्यंजक ज्ञात करना आदि प्रमुख है। इस प्रकार के प्रश्नो को हल करने के लिए मूल अवधारणाओं की जानकारी होना आवश्यक है।

वर्ग (Square)- किसी संख्या को स्वयं से ही गुणा करने पर प्राप्त संख्या दी गई संख्या का वर्ग कहलाती है।  जैसे 5 का वर्ग = 5×5 =25

वर्गमूल (Square Roots)- किसी संख्या का वर्गमूल वह संख्या है, जिसे अपने से ही गुणा करने पर दी गई संख्या प्राप्त होती है या किसी दी गई संख्या का वर्गमूल वह संख्या है जिसका वर्ग उस दी गई संख्या के बराबर है।  इसे ‘√’ चिन्ह से प्रदर्शित करते है।  जैसे √16 = √4 x 4 = 4 तथा 4 का वर्ग 4 x 4 = 16 होता है। 

वर्ग एवं वर्गमूल के सवाल एवं सूत्र – Square And Square Roots 

संयुग्मी की गुणा

जब कोई संख्या 1 / √a – √b के रूप मे दी गई हुई होती है तो उसमे विपरीत चिन्ह वाली संख्या अर्थात संयुग्मी के 1 / √a + √b की गुणा अंश तथा हल दोनों मे कर देते है।

Square And Square Roots Problems Solve in Hindi

याद रखने वाली योग्य बाते ….

• यदि किसी संख्या के अंको की संख्या सम ( माना m ) हो , तो उसके वर्गमूल मे अंको की संख्या [m / 2] होती है।
• यदि किसी संख्या के अंको की संख्या विषम ( माना n ) हो, तो उसके वर्गमूल मे अंको की संख्या [m+1 / 2] होती है।
• किसी संख्या का पूर्ण संख्यात्मक वर्गमूल तभी संभव है जब उस संख्या के इकाई अंक 0 , 1 , 4 , 5 या 9 मे कोई एक हो.

Fractions (भिन्न) I Types of Fraction

Fractions भिन्न

यदि किसी संख्या को p/q के रूप में जहाँ p और q पूर्णांक हैं तथा q ≠ 0 लिखा जाये तो ऐसी संख्या को भिन्न कहते हैं। भिन्न में भाज्य को एक रेखा के उपर तथा भाजक को रेखा के नीचे लिखा जाता है, ऊपर की संख्या अर्थात भाज्य को अंश तथा नीचे की संख्या अर्थात भाजक को हर कहा जाता है। 1/3, 4/5, 6/7आदि भिन्न के उदाहरण हैं जिसमें 1, 4 , 6 अंश तथा 3, 5, 7 हर हैं।

Fractions (भिन्न) and Types of Fraction

Types Of Fractions भिन्नों के प्रकार

उचित भिन्न: यदि भिन्न का अंश हर से कम हो, तो भिन्न को उचित भिन्न कहते हैं।

जैसे- 2/3, 5/6, 11/10 …. इत्यादि।

अनुचित भिन्न: यदि भिन्न का अंश हर से बड़ा हो तो भिन्न को अनुचित भिन्न कहते हैं।

जैसे- 2/3, 5/6, 8/7 …. इत्यादि।

मिश्र भिन्न: यदि भिन्न एक पूर्णांक तथा भिन्न से मिलकर बनी हो तो भिन्न को मिश्र भिन्न कहते हैं।

जैसे –     …. इत्यादि।

मिश्रित भिन्न: यदि अंश या हर या दोनों भिन्न हो, तो भिन्न को मिश्रित भिन्न कहते हैं।

जैसे-     …. इत्यादि।

दशमलव भिन्न: वे भिन्न जिनके हर 10, 10² या 10³ इत्यादि हो, तो दशमलव भिन्न कहलाते हैं।

जैसे-     …. इत्यादि।

वितत भिन्न: सामान्य तौर पर किसी भिन्न के हर या कभी-कभी अंश में किसी संख्या के जोड़ने या घटाने से बनने वाले भिन्न को वितत भिन्न कहते हैं।

जैसे-    …. इत्यादि।

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