Answer :
To find a quadratic equation in standard form with given roots [tex]\(\frac{5}{3}\)[/tex] and [tex]\(\frac{7}{3}\)[/tex], we can follow these steps:
1. Express the quadratic equation with roots: If a quadratic equation has roots [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex], it can be written in the form:
[tex]\[
(x - r_1)(x - r_2) = 0
\][/tex]
Given [tex]\(r_1 = \frac{5}{3}\)[/tex] and [tex]\(r_2 = \frac{7}{3}\)[/tex], we have:
[tex]\[
(x - \frac{5}{3})(x - \frac{7}{3}) = 0
\][/tex]
2. Expand the equation:
[tex]\[
\left(x - \frac{5}{3}\right)\left(x - \frac{7}{3}\right)
\][/tex]
Using the distributive property (FOIL method):
[tex]\[
x^2 - \frac{7}{3}x - \frac{5}{3}x + \left(\frac{5}{3} \times \frac{7}{3}\right) = 0
\][/tex]
Simplify:
[tex]\[
x^2 - \left(\frac{7}{3} + \frac{5}{3}\right)x + \frac{35}{9} = 0
\][/tex]
Combine the fractions:
[tex]\[
x^2 - \frac{12}{3}x + \frac{35}{9} = 0
\][/tex]
Simplify further:
[tex]\[
x^2 - 4x + \frac{35}{9} = 0
\][/tex]
3. Clear the fractions by multiplying through by the least common multiple (LCM):
The LCM of the denominator (which is 9) can be used to clear fractions:
[tex]\[
9 \left( x^2 - 4x + \frac{35}{9} \right) = 0
\][/tex]
Distribute 9:
[tex]\[
9x^2 - 36x + 35 = 0
\][/tex]
So, the quadratic equation in standard form with the roots [tex]\(\frac{5}{3}\)[/tex] and [tex]\(\frac{7}{3}\)[/tex] is:
[tex]\[
9x^2 - 36x + 35 = 0
\][/tex]
Comparing this with the options provided:
- A [tex]\(9x^2 - 36x + 35 = 0\)[/tex]
- B [tex]\(9x^2 + 36x - 35 = 0\)[/tex]
- C [tex]\(9x^2 + 36x + 35 = 0\)[/tex]
- D [tex]\(9x^2 - 36x - 35 = 0\)[/tex]
The correct answer is:
```
A [tex]\(9x^2 - 36x + 35 = 0\)[/tex]
```
1. Express the quadratic equation with roots: If a quadratic equation has roots [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex], it can be written in the form:
[tex]\[
(x - r_1)(x - r_2) = 0
\][/tex]
Given [tex]\(r_1 = \frac{5}{3}\)[/tex] and [tex]\(r_2 = \frac{7}{3}\)[/tex], we have:
[tex]\[
(x - \frac{5}{3})(x - \frac{7}{3}) = 0
\][/tex]
2. Expand the equation:
[tex]\[
\left(x - \frac{5}{3}\right)\left(x - \frac{7}{3}\right)
\][/tex]
Using the distributive property (FOIL method):
[tex]\[
x^2 - \frac{7}{3}x - \frac{5}{3}x + \left(\frac{5}{3} \times \frac{7}{3}\right) = 0
\][/tex]
Simplify:
[tex]\[
x^2 - \left(\frac{7}{3} + \frac{5}{3}\right)x + \frac{35}{9} = 0
\][/tex]
Combine the fractions:
[tex]\[
x^2 - \frac{12}{3}x + \frac{35}{9} = 0
\][/tex]
Simplify further:
[tex]\[
x^2 - 4x + \frac{35}{9} = 0
\][/tex]
3. Clear the fractions by multiplying through by the least common multiple (LCM):
The LCM of the denominator (which is 9) can be used to clear fractions:
[tex]\[
9 \left( x^2 - 4x + \frac{35}{9} \right) = 0
\][/tex]
Distribute 9:
[tex]\[
9x^2 - 36x + 35 = 0
\][/tex]
So, the quadratic equation in standard form with the roots [tex]\(\frac{5}{3}\)[/tex] and [tex]\(\frac{7}{3}\)[/tex] is:
[tex]\[
9x^2 - 36x + 35 = 0
\][/tex]
Comparing this with the options provided:
- A [tex]\(9x^2 - 36x + 35 = 0\)[/tex]
- B [tex]\(9x^2 + 36x - 35 = 0\)[/tex]
- C [tex]\(9x^2 + 36x + 35 = 0\)[/tex]
- D [tex]\(9x^2 - 36x - 35 = 0\)[/tex]
The correct answer is:
```
A [tex]\(9x^2 - 36x + 35 = 0\)[/tex]
```
