Construct a quadratic equation whose roots are -5 and -²/⁹

To construct a quadratic equation given its roots, you can use the fact that if \( p \) and \( q \) are the roots of a quadratic equation, then the quadratic equation can be expressed as:

\[
x^2 - (p + q)x + pq = 0
\]

In this case, the given roots are \( -5 \) and \( -\frac{2}{9} \).

1. **Sum of the roots (\( p + q \))**:  
   \[
   p + q = -5 + \left(-\frac{2}{9}\right) = -5 - \frac{2}{9}
   \]

   To combine these, we need a common denominator:

   \[
   -5 = -\frac{45}{9}
   \]

   Therefore:

   \[
   p + q = -\frac{45}{9} - \frac{2}{9} = -\frac{47}{9}
   \]

2. **Product of the roots (\( pq \))**:  
   \[
   pq = \left(-5\right) \left(-\frac{2}{9}\right) = \frac{10}{9}
   \]

Now, substitute these values into the quadratic equation form:

\[
x^2 - (p + q)x + pq = 0
\]

Substituting the values:

\[
x^2 - \left(-\frac{47}{9}\right)x + \frac{10}{9} = 0
\]

Simplifying this:

\[
x^2 + \frac{47}{9}x + \frac{10}{9} = 0
\]

To eliminate the fractions, multiply the entire equation by 9:

\[
9x^2 + 47x + 10 = 0
\]

This is the quadratic equation whose roots are \( -5 \) and \( -\frac{2}{9} \).

Would you like more details or have any questions?

Here are 5 related questions to explore further:

1. What are the coefficients of a quadratic equation, and how are they related to its roots?
2. How can you find the sum and product of the roots of a quadratic equation directly from its standard form?
3. Can a quadratic equation have complex roots? If so, what would the equation look like?
4. How do you determine the vertex of a parabola represented by a quadratic equation?
5. What are the methods for solving a quadratic equation, and how do they differ?

**Tip:** Remember, the discriminant of a quadratic equation \( ax^2 + bx + c = 0 \) can tell you about the nature of its roots: \( b^2 - 4ac \). If it's positive, the roots are real and distinct; if zero, the roots are real and identical; and if negative, the roots are complex.

Learn how to construct a quadratic equation with roots -5 and -2/9 using the sum and product of roots theorem. Step-by-step explanation with the quadratic equation formula and simplification. Suitable for high school students studying quadratic equations.

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