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]
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]
