quadratic equation

Published: 19 Jul 2025

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

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_k\right\},\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 ab\frac{a}{b} is ba\frac{b}{a}

Examples:

Reciprocal of 34\frac{3}{4} is 43\frac{4}{3}
Reciprocal of −57\frac{-5}{7} is 7−5\frac{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×dc\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}

Example: 57÷23=57×32=1514\frac{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=158\frac{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=215\frac{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=415\frac{2}{5} \div 1\frac{1}{2} = \frac{2}{5} \times \frac{2}{3} = \frac{4}{15}

🔸 6. Mixed Number ÷ Fraction

213÷12=73×21=1432\frac{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=97\frac{5}{7} \div \frac{5}{9} = \frac{9}{7}

🔸 9. Fractions with Same Denominators

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

🔸 10. Same Rational Numbers

78÷78=1\frac{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=undefined\frac{3}{4} \div 0 = \text{undefined}

🔸 13. Dividing by Reciprocal

23÷32=23×23=49\frac{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= ?1\frac{1}{2} \div \frac{1}{3} = \, ?
56÷56= ?\frac{5}{6} \div \frac{5}{6} = \, ?
25÷123= ?\frac{2}{5} \div 1\frac{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
✅ Want it in Urdu or another language?

Let me know if you’d like:

A PDF version of this with practice problems
A LaTeX copy for print
A custom HTML version for WordPress
A worksheet with answers

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

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

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