Add and Subtract Rational Expressions with a Common Denominator

Add Rational Expressions with a Common Denominator

What is the first step you take when you add numerical fractions? You check if they have a common denominator. If they do, you add the numerators and place the sum over the common denominator. If they do not have a common denominator, you find one before you add.

It is the same with rational expressions. To add rational expressions, they must have a common denominator. When the denominators are the same, you add the numerators and place the sum over the common denominator.

We will add two numerical fractions first, to remind us of how this is done.

Example

Try it.

Add: \(\frac{5}{18}+\frac{7}{18}.\)

Solution

	\(\frac{5}{18}+\frac{7}{18}\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{5+7}{18}\)
Add in the numerator.	\(\frac{12}{18}\)
Factor the numerator and denominator to show the common factors.	\(\frac{6\cdot 2}{6\cdot 3}\)
Remove common factors.	\(\frac{6\cdot 2}{6\cdot 3}\)
Simplify.	\(\frac{2}{3}\)

Remember, we do not allow values that would make the denominator zero. What value of \(y\) should be excluded in the next example?

Example

Try it.

Add: \(\frac{3y}{4y-3}+\frac{7}{4y-3}.\)

Solution

	\(\frac{3y}{4y-3}+\frac{7}{4y-3}\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{3y+7}{4y-3}\)

The numerator and denominator cannot be factored. The fraction is simplified.

Example

Try it.

Add: \(\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}.\)

Solution

	\(\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}\)
The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{7x+12+{x}^{2}}{x+3}\)
Write the degrees in descending order.	\(\frac{{x}^{2}+7x+12}{x+3}\)
Factor the numerator.	\(\frac{(x+3)(x+4)}{x+3}\)
Simplify by removing common factors.	\(\frac{(x+3)(x+4)}{x+3}\)
Simplify.	\(x+4\)

Subtract Rational Expressions with a Common Denominator

To subtract rational expressions, they must also have a common denominator. When the denominators are the same, you subtract the numerators and place the difference over the common denominator.

We always simplify rational expressions. Be sure to factor, if possible, after you subtract the numerators so you can identify any common factors.

Example

Try it.

Subtract: \(\frac{{n}^{2}}{n-10}-\frac{100}{n-10}.\)

Solution

	\(\frac{{n}^{2}}{n-10}-\frac{100}{n-10}\)
The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator.	\(\frac{{n}^{2}-100}{n-10}\)
Factor the numerator.	\(\frac{(n-10)(n+10)}{n-10}\)
Simplify by removing common factors.	\(\frac{(n-10)(n+10)}{n-10}\)
Simplify.	\(n+10\)

Be careful of the signs when you subtract a binomial!

Example

Try it.

Subtract: \(\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}.\)

Solution

	\(\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}\)
The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator.	\(\frac{{y}^{2}-(2y+24)}{y-6}\)
Distribute the sign in the numerator.	\(\frac{{y}^{2}-2y-24}{y-6}\)
Factor the numerator.	\(\frac{(y-6)(y+4)}{y-6}\)
Remove common factors.	\(\frac{(y-6)(y+4)}{y-6}\)
Simplify.	\(y+4\)

Example

Try it.

Subtract: \(\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}.\)

Solution

	\(\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}\)
Subtract the numerators and place the difference over the common denominator.	\(\frac{5{x}^{2}-7x+3-(4{x}^{2}+x-9)}{{x}^{2}-3x-18}\)
Distribute the sign in the numerator.	\(\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}\)
Combine like terms.	\(\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}\)
Factor the numerator and the denominator.	\(\frac{(x-2)(x-6)}{(x+3)(x-6)}\)
Simplify by removing common factors.	\(\frac{(x-2)(x-6)}{(x+3)(x-6)}\)
Simplify.	\(\frac{(x-2)}{(x+3)}\)

Add and Subtract Rational Expressions whose Denominators are Opposites

When the denominators of two rational expressions are opposites, it is easy to get a common denominator. We just have to multiply one of the fractions by \(\frac{-1}{-1}\).

Let’s see how this works.

Multiply the second fraction by \(\frac{-1}{-1}\).
The denominators are the same.
Simplify.

Example

Try it.

Add: \(\frac{4u-1}{3u-1}+\frac{u}{1-3u}.\)

Solution

The denominators are opposites, so multiply the second fraction by \(\frac{-1}{-1}\).
Simplify the second fraction.
The denominators are the same. Add the numerators.
Simplify.
Simplify.

Example

Try it.

Subtract: \(\frac{{m}^{2}-6m}{{m}^{2}-1}-\frac{3m+2}{1-{m}^{2}}.\)

Solution

The denominators are opposites, so multiply the second fraction by \(\frac{-1}{-1}\).
Simplify the second fraction.
The denominators are the same. Subtract the numerators.
Distribute.	\(\frac{{m}^{2}-6m+3m+2}{{m}^{2}-1}\)
Combine like terms.
Factor the numerator and denominator.
Simplify by removing common factors.
Simplify.

Key Concepts

• Rational Expression Addition
  • If \(p,q,\ \text{and}\ r\) are polynomials where \(r\ne 0\), then
    \[\frac{p}{r}+\frac{q}{r}=\frac{p+q}{r}\]
  • To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
• Rational Expression Subtraction
  • If \(p,q,\ \text{and}\ r\) are polynomials where \(r\ne 0\), then
    \[\frac{p}{r}-\frac{q}{r}=\frac{p-q}{r}\]
  • To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

Add and Subtract Rational Expressions with a Common Denominator

Add Rational Expressions with a Common Denominator

In the following exercises, add.

Try it.

\(\frac{2}{15}+\frac{7}{15}\)

Solution

\(\frac{3}{5}\)

Try it.

\(\frac{4}{21}+\frac{3}{21}\)

Try it.

\(\frac{7}{24}+\frac{11}{24}\)

Solution

\(\frac{3}{4}\)

Try it.

\(\frac{7}{36}+\frac{13}{36}\)

Try it.

\(\frac{3a}{a-b}+\frac{1}{a-b}\)

Solution

\(\frac{3a+1}{a-b}\)

Try it.

\(\frac{3c}{4c-5}+\frac{5}{4c-5}\)

Try it.

\(\frac{d}{d+8}+\frac{5}{d+8}\)

Solution

\(\frac{d+5}{d+8}\)

Try it.

\(\frac{7m}{2m+n}+\frac{4}{2m+n}\)

Try it.

\(\frac{{p}^{2}+10p}{p+2}+\frac{16}{p+2}\)

Solution

\(p+8\)

Try it.

\(\frac{{q}^{2}+12q}{q+3}+\frac{27}{q+3}\)

Try it.

\(\frac{2{r}^{2}}{2r-1}+\frac{15r-8}{2r-1}\)

Solution

\(r+8\)

Try it.

\(\frac{3{s}^{2}}{3s-2}+\frac{13s-10}{3s-2}\)

Try it.

\(\frac{8{t}^{2}}{t+4}+\frac{32t}{t+4}\)

Solution

\(8t\)

Try it.

\(\frac{6{v}^{2}}{v+5}+\frac{30v}{v+5}\)

Try it.

\(\frac{2{w}^{2}}{{w}^{2}-16}+\frac{8w}{{w}^{2}-16}\)

Solution

\(\frac{2w}{w-4}\)

Try it.

\(\frac{7{x}^{2}}{{x}^{2}-9}+\frac{21x}{{x}^{2}-9}\)

Subtract Rational Expressions with a Common Denominator

In the following exercises, subtract.

Try it.

\(\frac{{y}^{2}}{y+8}-\frac{64}{y+8}\)

Solution

\(y-8\)

Try it.

\(\frac{{z}^{2}}{z+2}-\frac{4}{z+2}\)

Try it.

\(\frac{9{a}^{2}}{3a-7}-\frac{49}{3a-7}\)

Solution

\(3a+7\)

Try it.

\(\frac{25{b}^{2}}{5b-6}-\frac{36}{5b-6}\)

Try it.

\(\frac{{c}^{2}}{c-8}-\frac{6c+16}{c-8}\)

Solution

\(c+2\)

Try it.

\(\frac{{d}^{2}}{d-9}-\frac{6d+27}{d-9}\)

Try it.

\(\frac{3{m}^{2}}{6m-30}-\frac{21m-30}{6m-30}\)

Solution

\(\frac{m-2}{2}\)

Try it.

\(\frac{2{n}^{2}}{4n-32}-\frac{18n-16}{4n-32}\)

Try it.

\(\frac{6{p}^{2}+3p+4}{{p}^{2}+4p-5}-\frac{5{p}^{2}+p+7}{{p}^{2}+4p-5}\)

Solution

\(\frac{p+3}{p+5}\)

Try it.

\(\frac{5{q}^{2}+3q-9}{{q}^{2}+6q+8}-\frac{4{q}^{2}+9q+7}{{q}^{2}+6q+8}\)

Try it.

\(\frac{5{r}^{2}+7r-33}{{r}^{2}-49}-\frac{4{r}^{2}+5r+30}{{r}^{2}-49}\)

Solution

\(\frac{r+9}{r+7}\)

Try it.

\(\frac{7{t}^{2}-t-4}{{t}^{2}-25}-\frac{6{t}^{2}+12t-44}{{t}^{2}-25}\)

Add and Subtract Rational Expressions whose Denominators are Opposites

In the following exercises, add.

Try it.

\(\frac{10v}{2v-1}+\frac{2v+4}{1-2v}\)

Solution

\(4\)

Try it.

\(\frac{20w}{5w-2}+\frac{5w+6}{2-5w}\)

Try it.

\(\frac{10{x}^{2}+16x-7}{8x-3}+\frac{2{x}^{2}+3x-1}{3-8x}\)

Solution

\(x+2\)

Try it.

\(\frac{6{y}^{2}+2y-11}{3y-7}+\frac{3{y}^{2}-3y+17}{7-3y}\)

In the following exercises, subtract.

Try it.

\(\frac{{z}^{2}+6z}{{z}^{2}-25}-\frac{3z+20}{25-{z}^{2}}\)

Solution

\(\frac{z+4}{z-5}\)

Try it.

\(\frac{{a}^{2}+3a}{{a}^{2}-9}-\frac{3a-27}{9-{a}^{2}}\)

Try it.

\(\frac{2{b}^{2}+30b-13}{{b}^{2}-49}-\frac{2{b}^{2}-5b-8}{49-{b}^{2}}\)

Solution

\(\frac{4b-3}{b-7}\)

Try it.

\(\frac{{c}^{2}+5c-10}{{c}^{2}-16}-\frac{{c}^{2}-8c-10}{16-{c}^{2}}\)

Try it.

Donald thinks that \(\frac{3}{x}+\frac{4}{x}\) is \(\frac{7}{2x}.\) Is Donald correct? Explain.

Try it.

Explain how you find the Least Common Denominator of \({x}^{2}+5x+4\) and \({x}^{2}-16.\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Haddaad Ma jiro xisaabis oo ku nool midkan, laakiin qaybaha ka mid ah waa la xisaabi karaa. Ku raaxayso mid ka hooseeya, ama ku qor adiga kuu gaar ah.

⌨ Nooc ✎ Sawir

Su'aalkaaga xisaabta

π Xalin

Tababar (40)

Ku day mid kasta oo ka mid ah oo ku saabsan buugaag hore. Muuji jawaabta si aad u hubiso; kuwa la xaqiijiyay waxaa la furi karaa in xalka ah tallaabo kasta.

1. Add: \(\frac{y}{3}+\frac{9}{3}.\)

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(\frac{y+9}{3}\)

2. Subtract: \(\frac{10}{x}-\frac{2}{x}.\)

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(\frac{8}{x}\)

3. Factor completely: \(8{n}^{5}-20{n}^{3}.\)

   Shaqada iyada oo Tallaabo

4. Factor completely: \(45{a}^{3}-5a{b}^{2}.\)

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(5a\left(3a-b\right)\left(3a+b\right)\)

5. Add: \(\frac{5}{18}+\frac{7}{18}.\)

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   	\(\frac{5}{18}+\frac{7}{18}\)
   The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{5+7}{18}\)
   Add in the numerator.	\(\frac{12}{18}\)
   Factor the numerator and denominator to show the common factors.	\(\frac{6\cdot 2}{6\cdot 3}\)
   Remove common factors.	\(\frac{6\cdot 2}{6\cdot 3}\)
   Simplify.	\(\frac{2}{3}\)

6. Add: \(\frac{7}{16}+\frac{5}{16}.\)

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(\frac{3}{4}\)

7. Add: \(\frac{3}{10}+\frac{1}{10}.\)

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(\frac{2}{5}\)

8. Add: \(\frac{3y}{4y-3}+\frac{7}{4y-3}.\)

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   	\(\frac{3y}{4y-3}+\frac{7}{4y-3}\)
   The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{3y+7}{4y-3}\)

   The numerator and denominator cannot be factored. The fraction is simplified.

9. Add: \(\frac{5x}{2x+3}+\frac{2}{2x+3}.\)

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(\frac{5x+2}{2x+3}.\)

10. Add: \(\frac{x}{x-2}+\frac{1}{x-2}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(\frac{x+1}{x-2}\)

11. Add: \(\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    	\(\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}\)
    The fractions have a common denominator, so add the numerators and place the sum over the common denominator.	\(\frac{7x+12+{x}^{2}}{x+3}\)
    Write the degrees in descending order.	\(\frac{{x}^{2}+7x+12}{x+3}\)
    Factor the numerator.	\(\frac{(x+3)(x+4)}{x+3}\)
    Simplify by removing common factors.	\(\frac{(x+3)(x+4)}{x+3}\)
    Simplify.	\(x+4\)

12. Add: \(\frac{9x+14}{x+7}+\frac{{x}^{2}}{x+7}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(x+2\)

13. Add: \(\frac{{x}^{2}+8x}{x+5}+\frac{15}{x+5}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(x+3\)

14. Subtract: \(\frac{{n}^{2}}{n-10}-\frac{100}{n-10}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    	\(\frac{{n}^{2}}{n-10}-\frac{100}{n-10}\)
    The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator.	\(\frac{{n}^{2}-100}{n-10}\)
    Factor the numerator.	\(\frac{(n-10)(n+10)}{n-10}\)
    Simplify by removing common factors.	\(\frac{(n-10)(n+10)}{n-10}\)
    Simplify.	\(n+10\)

15. Subtract: \(\frac{{x}^{2}}{x+3}-\frac{9}{x+3}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(x-3\)

16. Subtract: \(\frac{4{x}^{2}}{2x-5}-\frac{25}{2x-5}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(2x+5\)

17. Subtract: \(\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    	\(\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}\)
    The fractions have a common denominator, so subtract the numerators and place the difference over the common denominator.	\(\frac{{y}^{2}-(2y+24)}{y-6}\)
    Distribute the sign in the numerator.	\(\frac{{y}^{2}-2y-24}{y-6}\)
    Factor the numerator.	\(\frac{(y-6)(y+4)}{y-6}\)
    Remove common factors.	\(\frac{(y-6)(y+4)}{y-6}\)
    Simplify.	\(y+4\)

18. Subtract: \(\frac{{n}^{2}}{n-4}-\frac{n+12}{n-4}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(n+3\)

19. Subtract: \(\frac{{y}^{2}}{y-1}-\frac{9y-8}{y-1}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(y-8\)

20. Subtract: \(\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    	\(\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}\)
    Subtract the numerators and place the difference over the common denominator.	\(\frac{5{x}^{2}-7x+3-(4{x}^{2}+x-9)}{{x}^{2}-3x-18}\)
    Distribute the sign in the numerator.	\(\frac{5{x}^{2}-7x+3-4{x}^{2}-x+9}{{x}^{2}-3x-18}\)
    Combine like terms.	\(\frac{{x}^{2}-8x+12}{{x}^{2}-3x-18}\)
    Factor the numerator and the denominator.	\(\frac{(x-2)(x-6)}{(x+3)(x-6)}\)
    Simplify by removing common factors.	\(\frac{(x-2)(x-6)}{(x+3)(x-6)}\)
    Simplify.	\(\frac{(x-2)}{(x+3)}\)

21. Subtract: \(\frac{4{x}^{2}-11x+8}{{x}^{2}-3x+2}-\frac{3{x}^{2}+x-3}{{x}^{2}-3x+2}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(\frac{x-11}{x-2}\)

22. Subtract: \(\frac{6{x}^{2}-x+20}{{x}^{2}-81}-\frac{5{x}^{2}+11x-7}{{x}^{2}-81}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(\frac{x-3}{x+9}\)

23. Add: \(\frac{4u-1}{3u-1}+\frac{u}{1-3u}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    The denominators are opposites, so multiply the second fraction by \(\frac{-1}{-1}\).
    Simplify the second fraction.
    The denominators are the same. Add the numerators.
    Simplify.
    Simplify.

24. Add: \(\frac{8x-15}{2x-5}+\frac{2x}{5-2x}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(3\)

25. Add: \(\frac{6{y}^{2}+7y-10}{4y-7}+\frac{2{y}^{2}+2y+11}{7-4y}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(y+3\)

26. Subtract: \(\frac{{m}^{2}-6m}{{m}^{2}-1}-\frac{3m+2}{1-{m}^{2}}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    The denominators are opposites, so multiply the second fraction by \(\frac{-1}{-1}\).
    Simplify the second fraction.
    The denominators are the same. Subtract the numerators.
    Distribute.	\(\frac{{m}^{2}-6m+3m+2}{{m}^{2}-1}\)
    Combine like terms.
    Factor the numerator and denominator.
    Simplify by removing common factors.
    Simplify.

27. Subtract: \(\frac{{y}^{2}-5y}{{y}^{2}-4}-\frac{6y-6}{4-{y}^{2}}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(\frac{y+3}{y+2}\)

28. Subtract: \(\frac{2{n}^{2}+8n-1}{{n}^{2}-1}-\frac{{n}^{2}-7n-1}{1-{n}^{2}}.\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(\frac{3n-2}{n-1}\)

29. \(\frac{2}{15}+\frac{7}{15}\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(\frac{3}{5}\)

30. \(\frac{4}{21}+\frac{3}{21}\)

    Shaqada iyada oo Tallaabo

31. \(\frac{7}{24}+\frac{11}{24}\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(\frac{3}{4}\)

32. \(\frac{7}{36}+\frac{13}{36}\)

    Shaqada iyada oo Tallaabo

33. \(\frac{3a}{a-b}+\frac{1}{a-b}\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(\frac{3a+1}{a-b}\)

34. \(\frac{3c}{4c-5}+\frac{5}{4c-5}\)

    Shaqada iyada oo Tallaabo

35. \(\frac{d}{d+8}+\frac{5}{d+8}\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(\frac{d+5}{d+8}\)

36. \(\frac{7m}{2m+n}+\frac{4}{2m+n}\)

    Shaqada iyada oo Tallaabo

37. \(\frac{{p}^{2}+10p}{p+2}+\frac{16}{p+2}\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(p+8\)

38. \(\frac{{q}^{2}+12q}{q+3}+\frac{27}{q+3}\)

    Shaqada iyada oo Tallaabo

39. \(\frac{2{r}^{2}}{2r-1}+\frac{15r-8}{2r-1}\)

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(r+8\)

40. \(\frac{3{s}^{2}}{3s-2}+\frac{13s-10}{3s-2}\)

    Shaqada iyada oo Tallaabo

Ku hay shaqadaada

A xisaab bilaash ah oo ku daray qoraalada waxbarasho kasta, diiwaanka waxa aad dhamaatay, dhibaatooyinka aad xal u tahay meel kaliya, iyo macallinka aad weydiisan kartaa oo ku saabsan bogga. Xisaabta oo keliya waa u furan yahay qof kasta, la diiwaan geliyey ama aan.

Ku soo biir Ku soo gal

Calaamadaha la isticmaalay halkan

Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

f'(x)derivative
Instantaneous rate of change; slope of the graph.\neqnot equal
The two sides are different.\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Sida loo: Add and Subtract Rational Expressions with a Common Denominator

1. Add rational expressions with a common denominator
2. Subtract rational expressions with a common denominator
3. Add and subtract rational expressions whose denominators are opposites
4. If
5. To add rational expressions with a common denominator, add the numerators and place the sum over the common denominator.
6. If
7. To subtract rational expressions, subtract the numerators and place the difference over the common denominator.

Su'aalaha dadku waydiiyaan

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Qaybaha ee boggaan waxaa la beddelay ka OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed iyo dib-oo halkan lagu sharaxay; khaladaadka waa inoo. Sababtoo ah asalka waxaa loo oggolaaday in aan loo isticmaalin ganacsi, fasalkan waa bilaash, waxba ma iibiyo, iyo waxaa lagu wadaagayaa liisan la mid ah.

In ka badan Algebra

Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value