High School

Dear beloved readers, welcome to our website! We hope your visit here brings you valuable insights and meaningful inspiration. Thank you for taking the time to stop by and explore the content we've prepared for you.
------------------------------------------------ Write a quadratic equation in standard form that has [tex]\(\frac{5}{3}\)[/tex] and [tex]\(\frac{7}{3}\)[/tex] as its roots.

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]

Answer :

To find the quadratic equation in standard form with given roots [tex]\(\frac{5}{3}\)[/tex] and [tex]\(\frac{7}{3}\)[/tex], follow these steps:

1. Understand the form of a quadratic equation:
A quadratic equation in standard form is written as [tex]\(ax^2 + bx + c = 0\)[/tex].

2. Use the fact that roots can form a quadratic equation:
If a quadratic equation has roots [tex]\(r_1\)[/tex] and [tex]\(r_2\)[/tex], it can be expressed as:
[tex]\[ a(x - r_1)(x - r_2) = 0 \][/tex]

3. Identify the roots:
The roots given are [tex]\(\frac{5}{3}\)[/tex] and [tex]\(\frac{7}{3}\)[/tex].

4. Find the sum and product of the roots:
- The sum of the roots [tex]\(r_1 + r_2 = \frac{5}{3} + \frac{7}{3} = \frac{12}{3} = 4\)[/tex].
- The product of the roots [tex]\(r_1 \times r_2 = \frac{5}{3} \times \frac{7}{3} = \frac{35}{9}\)[/tex].

5. Write the quadratic equation:
Use the formula for the quadratic equation derived from its roots:
[tex]\[ a(x^2 - (r_1 + r_2)x + r_1r_2) = 0 \][/tex]
Plug in the values:
[tex]\[ a(x^2 - 4x + \frac{35}{9}) = 0 \][/tex]

6. Clear fractions to find integer coefficients:
To avoid fractions, choose [tex]\(a = 9\)[/tex] to make the equation have integer coefficients. So, multiply everything by 9:
[tex]\[ 9(x^2 - 4x + \frac{35}{9}) = 0 \][/tex]
Simplifying this:
[tex]\[ 9x^2 - 36x + 35 = 0 \][/tex]

So, the quadratic equation in standard form is [tex]\(9x^2 - 36x + 35 = 0\)[/tex].

The correct option is:

A) [tex]\(9x^2 - 36x + 35 = 0\)[/tex]