Find the best solutions to your questions at Westonci.ca, the premier Q&A platform with a community of knowledgeable experts. Ask your questions and receive accurate answers from professionals with extensive experience in various fields on our platform. Explore comprehensive solutions to your questions from knowledgeable professionals across various fields on our platform.
Sagot :
To construct a quadratic equation given its roots, we need to use the fact that for a quadratic equation of the form [tex]\( ax^2 + bx + c = 0 \)[/tex], if [tex]\(\alpha\)[/tex] and [tex]\(\beta\)[/tex] are the roots, then:
1. The sum of the roots, [tex]\(\alpha + \beta = -\frac{b}{a}\)[/tex]
2. The product of the roots, [tex]\(\alpha \beta = \frac{c}{a}\)[/tex]
Given the roots [tex]\(\frac{4}{5}\)[/tex] and [tex]\(-2 \frac{2}{3}\)[/tex], we first convert these numbers to a common fraction form:
- [tex]\(\frac{4}{5}\)[/tex] is already in fraction form.
- [tex]\(-2 \frac{2}{3}\)[/tex] can be converted to an improper fraction:
[tex]\[ -2 \frac{2}{3} = -\left(2 + \frac{2}{3}\right) = -\left(\frac{6}{3} + \frac{2}{3}\right) = -\frac{8}{3} \][/tex]
Now, let's denote the roots as:
[tex]\[ \alpha = \frac{4}{5}, \quad \beta = -\frac{8}{3} \][/tex]
Step 1: Calculate the sum of the roots.
Sum of the roots [tex]\(\alpha + \beta\)[/tex] is:
[tex]\[ \alpha + \beta = \frac{4}{5} + \left( -\frac{8}{3} \right) \][/tex]
To add these fractions, we need a common denominator. The least common multiple of 5 and 3 is 15. Converting both fractions to have this common denominator:
[tex]\[ \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}, \quad -\frac{8}{3} = \frac{-8 \times 5}{3 \times 5} = \frac{-40}{15} \][/tex]
Now we can add them:
[tex]\[ \frac{12}{15} + \frac{-40}{15} = \frac{12 - 40}{15} = \frac{-28}{15} \][/tex]
Step 2: Calculate the product of the roots.
Product of the roots [tex]\(\alpha \beta\)[/tex] is:
[tex]\[ \alpha \beta = \left(\frac{4}{5}\right) \left(-\frac{8}{3}\right) \][/tex]
Multiplying the fractions:
[tex]\[ \left(\frac{4}{5}\right) \left(-\frac{8}{3}\right) = \frac{4 \times (-8)}{5 \times 3} = \frac{-32}{15} \][/tex]
Step 3: Write the quadratic equation using the sum and product of the roots.
In general, the quadratic equation based on the sum and product of the roots is given by:
[tex]\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \][/tex]
Plugging in our values:
[tex]\[ x^2 - \left(\frac{-28}{15}\right)x + \left(\frac{-32}{15}\right) = 0 \][/tex]
Simplifying the signs:
[tex]\[ x^2 + \frac{28}{15}x - \frac{32}{15} = 0 \][/tex]
Thus, the quadratic equation whose roots are [tex]\(\frac{4}{5}\)[/tex] and [tex]\(-2\frac{2}{3}\)[/tex] is:
[tex]\[ x^2 + \frac{28}{15}x - \frac{32}{15} = 0 \][/tex]
1. The sum of the roots, [tex]\(\alpha + \beta = -\frac{b}{a}\)[/tex]
2. The product of the roots, [tex]\(\alpha \beta = \frac{c}{a}\)[/tex]
Given the roots [tex]\(\frac{4}{5}\)[/tex] and [tex]\(-2 \frac{2}{3}\)[/tex], we first convert these numbers to a common fraction form:
- [tex]\(\frac{4}{5}\)[/tex] is already in fraction form.
- [tex]\(-2 \frac{2}{3}\)[/tex] can be converted to an improper fraction:
[tex]\[ -2 \frac{2}{3} = -\left(2 + \frac{2}{3}\right) = -\left(\frac{6}{3} + \frac{2}{3}\right) = -\frac{8}{3} \][/tex]
Now, let's denote the roots as:
[tex]\[ \alpha = \frac{4}{5}, \quad \beta = -\frac{8}{3} \][/tex]
Step 1: Calculate the sum of the roots.
Sum of the roots [tex]\(\alpha + \beta\)[/tex] is:
[tex]\[ \alpha + \beta = \frac{4}{5} + \left( -\frac{8}{3} \right) \][/tex]
To add these fractions, we need a common denominator. The least common multiple of 5 and 3 is 15. Converting both fractions to have this common denominator:
[tex]\[ \frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}, \quad -\frac{8}{3} = \frac{-8 \times 5}{3 \times 5} = \frac{-40}{15} \][/tex]
Now we can add them:
[tex]\[ \frac{12}{15} + \frac{-40}{15} = \frac{12 - 40}{15} = \frac{-28}{15} \][/tex]
Step 2: Calculate the product of the roots.
Product of the roots [tex]\(\alpha \beta\)[/tex] is:
[tex]\[ \alpha \beta = \left(\frac{4}{5}\right) \left(-\frac{8}{3}\right) \][/tex]
Multiplying the fractions:
[tex]\[ \left(\frac{4}{5}\right) \left(-\frac{8}{3}\right) = \frac{4 \times (-8)}{5 \times 3} = \frac{-32}{15} \][/tex]
Step 3: Write the quadratic equation using the sum and product of the roots.
In general, the quadratic equation based on the sum and product of the roots is given by:
[tex]\[ x^2 - (\text{sum of roots})x + (\text{product of roots}) = 0 \][/tex]
Plugging in our values:
[tex]\[ x^2 - \left(\frac{-28}{15}\right)x + \left(\frac{-32}{15}\right) = 0 \][/tex]
Simplifying the signs:
[tex]\[ x^2 + \frac{28}{15}x - \frac{32}{15} = 0 \][/tex]
Thus, the quadratic equation whose roots are [tex]\(\frac{4}{5}\)[/tex] and [tex]\(-2\frac{2}{3}\)[/tex] is:
[tex]\[ x^2 + \frac{28}{15}x - \frac{32}{15} = 0 \][/tex]
Thanks for using our service. We're always here to provide accurate and up-to-date answers to all your queries. Your visit means a lot to us. Don't hesitate to return for more reliable answers to any questions you may have. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.
