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.

Indi sen Hiç bir hasapçy buny çözmez, emma onuň bölekleri hasaplanmak üçindir. Aşakda birini synla ýa-da özüňi ýaz.

⌨ _Täze ✎ Ony çyz

Siziň rime problemaňyz

π Çöl

Tälim (40)

Her birini kagyzda synlap gör. Çözgüni barlap görmek üçin aç; barlap görlenler her bir ädim üçin çözgüççide açylyp bilner.

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

   _Işle Eşik

   Jawaby görkez

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

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

   _Işle Eşik

   Jawaby görkez

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

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

   _Işle Eşik

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

   _Işle Eşik

   Jawaby görkez

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

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

   _Işle Eşik

   Jawaby görkez

   	\(\frac{5}{18}+\frac{7}{18}\)
   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: \(\frac{7}{16}+\frac{5}{16}.\)

   _Işle Eşik

   Jawaby görkez

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

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

   _Işle Eşik

   Jawaby görkez

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

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

   _Işle Eşik

   Jawaby görkez

   	\(\frac{3y}{4y-3}+\frac{7}{4y-3}\)
   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: \(\frac{5x}{2x+3}+\frac{2}{2x+3}.\)

   _Işle Eşik

   Jawaby görkez

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

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

    _Işle Eşik

    Jawaby görkez

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

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

    _Işle Eşik

    Jawaby görkez

    	\(\frac{7x+12}{x+3}+\frac{{x}^{2}}{x+3}\)
    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: \(\frac{9x+14}{x+7}+\frac{{x}^{2}}{x+7}.\)

    _Işle Eşik

    Jawaby görkez

    \(x+2\)

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

    _Işle Eşik

    Jawaby görkez

    \(x+3\)

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

    _Işle Eşik

    Jawaby görkez

    	\(\frac{{n}^{2}}{n-10}-\frac{100}{n-10}\)
    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: \(\frac{{x}^{2}}{x+3}-\frac{9}{x+3}.\)

    _Işle Eşik

    Jawaby görkez

    \(x-3\)

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

    _Işle Eşik

    Jawaby görkez

    \(2x+5\)

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

    _Işle Eşik

    Jawaby görkez

    	\(\frac{{y}^{2}}{y-6}-\frac{2y+24}{y-6}\)
    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: \(\frac{{n}^{2}}{n-4}-\frac{n+12}{n-4}.\)

    _Işle Eşik

    Jawaby görkez

    \(n+3\)

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

    _Işle Eşik

    Jawaby görkez

    \(y-8\)

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

    _Işle Eşik

    Jawaby görkez

    	\(\frac{5{x}^{2}-7x+3}{{x}^{2}-3x-18}-\frac{4{x}^{2}+x-9}{{x}^{2}-3x-18}\)
    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: \(\frac{4{x}^{2}-11x+8}{{x}^{2}-3x+2}-\frac{3{x}^{2}+x-3}{{x}^{2}-3x+2}.\)

    _Işle Eşik

    Jawaby görkez

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

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

    _Işle Eşik

    Jawaby görkez

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

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

    _Işle Eşik

    Jawaby görkez

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

    _Işle Eşik

    Jawaby görkez

    \(3\)

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

    _Işle Eşik

    Jawaby görkez

    \(y+3\)

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

    _Işle Eşik

    Jawaby görkez

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

    _Işle Eşik

    Jawaby görkez

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

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

    _Işle Eşik

    Jawaby görkez

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

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

    _Işle Eşik

    Jawaby görkez

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

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

    _Işle Eşik

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

    _Işle Eşik

    Jawaby görkez

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

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

    _Işle Eşik

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

    _Işle Eşik

    Jawaby görkez

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

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

    _Işle Eşik

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

    _Işle Eşik

    Jawaby görkez

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

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

    _Işle Eşik

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

    _Işle Eşik

    Jawaby görkez

    \(p+8\)

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

    _Işle Eşik

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

    _Işle Eşik

    Jawaby görkez

    \(r+8\)

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

    _Işle Eşik

Özüň işle

Mugt hasap her bir ders üçin notlar goşýar, tamamlanan zadyňyzyň ýazgysyny, çözen problemaňyzy bir ýerde, we bu sahypa barada sorap biljek mugallymy. Matematika özüçe her kim üçin açyk, girseň ýa girmeseň.

_Gün _Giriş

Şu ýerde ulanylan simwollar

Eň täze düşündirme, surat, we her harpyň manysy üçin her bir simwoly basyň.

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.

Nädip: 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.

Adamlar soraýan soraglar

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.

Bu sahypanyň käbir bölegi şundan alyndy OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Bu ýerde gysgaldylan we tekrar beýan edilen; hatalar biziňkidir. Bu dersiň çeşmesi kommersiýa däl ulanmak üçin lisenziyalanandygy üçin bu ders mugt, hiç zat satmaýar we şol bir lisenziya astynda paýlaşylýar.

_Ýaşa Algebra

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