quadratic equation

Published: 19 Jul 2025

[latexpage]
At first, we sample $f(x)$ in the $N$ ($N$ is odd) equidistant points around $x^*$:
[
f_k = f(x_k),: x_k = x^*+kh,: k=-frac{N-1}{2},dots,frac{N-1}{2}
]
[
begin{array}{r}4 5enclose{longdiv}{20}20 hline 0end{array}
]

where $h$ is some step.
Then we interpolate points ${(x_k,f_k)}$ by polynomial
begin{equation} label{eq:poly}
P_{N-1}(x)=sum_{j=0}^{N-1}{a_jx^j}
end{equation}
Its coefficients ${a_j}$ are found as a solution of system of linear equations:
begin{equation} label{eq:sys}
left{ P_{N-1}(x_k) = f_kright},quad k=-frac{N-1}{2},dots,frac{N-1}{2}
end{equation}
Here are references to existing equations: (ref{eq:poly}), (ref{eq:sys}).
Here is reference to non-existing equation (ref{eq:unknown}).

Great! Below is your improved, student-friendly, and WordPress-ready version of the blog on Division of Rational Numbers, with colored tips, simplified language, emojis, headings, and an optional quiz added for better engagement.

🧮 Division of Rational Numbers Made Easy!

👋 Introduction

Ever tried to share 3 pizzas among 4 friends? That’s where dividing rational numbers helps! Whether it’s cutting a cake, measuring ingredients, or solving real math problems—knowing how to divide rational numbers is super useful.

Let’s learn how to divide rational numbers using simple tricks, clear steps, and fun examples!

🔁 What is the Reciprocal of a Rational Number?

When we flip the numerator and denominator, we get the reciprocal of a rational number.

📌 General Rule:
The reciprocal of abfrac{a}{b} is bafrac{b}{a}

Examples:

Reciprocal of 34frac{3}{4} is 43frac{4}{3}
Reciprocal of −57frac{-5}{7} is 7−5frac{7}{-5}
Note: 00 has no reciprocal because we can’t divide by 0.

➗ Division of Rational Numbers – Basic Rule

To divide two rational numbers:

📌 Formula: ab÷cd=ab×dcfrac{a}{b} div frac{c}{d} = frac{a}{b} times frac{d}{c}

Example: 57÷23=57×32=1514frac{5}{7} div frac{2}{3} = frac{5}{7} times frac{3}{2} = frac{15}{14}

🥞 The KFC Method (Keep–Flip–Change)

This fun trick helps you divide fractions easily!

K – Keep the first fraction
F – Flip the second fraction (reciprocal)
C – Change division to multiplication

Example: 34÷25=34×52=158frac{3}{4} div frac{2}{5} = frac{3}{4} times frac{5}{2} = frac{15}{8}

🔢 Types of Division of Rational Numbers

🔸 1. Negative ÷ Negative

💡 Same signs give positive. −57÷−23=1514-frac{5}{7} div -frac{2}{3} = frac{15}{14}

🔸 2. Negative ÷ Positive

💡 Different signs give negative. −57÷23=−1514-frac{5}{7} div frac{2}{3} = frac{-15}{14}

🔸 3. Fraction ÷ Whole Number

💡 Turn the whole number into a fraction. 45÷6=45×16=215frac{4}{5} div 6 = frac{4}{5} times frac{1}{6} = frac{2}{15}

🔸 4. Whole Number ÷ Fraction

6÷23=61×32=96 div frac{2}{3} = frac{6}{1} times frac{3}{2} = 9

🔸 5. Fraction ÷ Mixed Number

💡 Convert mixed number to improper fraction. 25÷112=25×23=415frac{2}{5} div 1frac{1}{2} = frac{2}{5} times frac{2}{3} = frac{4}{15}

🔸 6. Mixed Number ÷ Fraction

213÷12=73×21=1432frac{1}{3} div frac{1}{2} = frac{7}{3} times frac{2}{1} = frac{14}{3}

🔸 7. Decimal ÷ Fraction

💡 Convert decimal to fraction. 0.6÷711=610×117=33350.6 div frac{7}{11} = frac{6}{10} times frac{11}{7} = frac{33}{35}

🔸 8. Fractions with Same Numerators

57÷59=97frac{5}{7} div frac{5}{9} = frac{9}{7}

🔸 9. Fractions with Same Denominators

49÷59=45frac{4}{9} div frac{5}{9} = frac{4}{5}

🔸 10. Same Rational Numbers

78÷78=1frac{7}{8} div frac{7}{8} = 1

🔸 11. Zero ÷ Rational Number

0÷56=00 div frac{5}{6} = 0

🔸 12. Rational Number ÷ 0

❌ Undefined 34÷0=undefinedfrac{3}{4} div 0 = text{undefined}

🔸 13. Dividing by Reciprocal

23÷32=23×23=49frac{2}{3} div frac{3}{2} = frac{2}{3} times frac{2}{3} = frac{4}{9}

✅ Quick Quiz for Practice

Try these before checking the answers!

35÷67= ?frac{3}{5} div frac{6}{7} = , ?
0÷29= ?0 div frac{2}{9} = , ?
112÷13= ?1frac{1}{2} div frac{1}{3} = , ?
56÷56= ?frac{5}{6} div frac{5}{6} = , ?
25÷123= ?frac{2}{5} div 1frac{2}{3} = , ?

📌 Conclusion

Dividing rational numbers becomes super easy if you:

✅ Convert all numbers to fractions
✅ Use the reciprocal
✅ Multiply and simplify
✅ Follow sign rules

💡 Remember:
Keep – Flip – Change
Practice daily, and you’ll master it!

🔗 Related Posts (Add Internal Links)

📄 Want More?

✅ Download a printable PDF version
✅ Need a practice worksheet? Let me know
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Let me know if you’d like:

A PDF version of this with practice problems
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I can generate those next. Just tell me what you want!

$$
begin{aligned}
frac{36}{4} times frac{7}{28}
& = frac{cancel{36}^{9}}{cancel{4}^{1}} times frac{cancel{7}^{1}}{cancel{28}^{4}}
&& text{👉 Cancel common factors (36 ÷ 4 = 9, 28 ÷ 7 = 4)} \
& = frac{9}{1} times frac{1}{4}
&& text{👉 Write the simplified fractions} \
& = frac{9 times 1}{1 times 4}
&& text{👉 Multiply numerators and denominators} \
& = frac{9}{4}
&& text{✅ Final answer}
end{aligned}
$$

$$
= frac{cancel{36}^{9}}{cancel{4}^{1}} times frac{cancel{7}^{1}}{cancel{28}^{4}}
$$

$$ = frac{cancel{36}^{9}}{cancel{4}^{1}} times frac{cancel{7}^{1}}{cancel{28}^{4}} $$

https://quizzory.in/id/686bff70f6eea4751f6f5fd9

[
=frac{2}{3}left(frac{3}{2}+frac{5}{3}right)=frac{2}{3}left(frac{9+10}{6}right)
]

[
begin{aligned}
d. (-77) + (+35) &= -77 + 35 \
&= -42 \
end{aligned}
]