Problem 43 - Algebra 2 and Trigonometry

Question

Write a quadratic equation with integer coefficients for each pair of roots. \(\frac{3}{2} i,-\frac{3}{2} i\)

Step-by-Step Solution

Answer

The quadratic equation is \( 4x^2 + 9 = 0 \).

Step 1: Understand the relation between roots and coefficients

A quadratic equation with roots \( p \) and \( q \) can be written in the form \( (x-p)(x-q) = 0 \). This expands to \( x^2 - (p+q)x + pq = 0 \). Since our roots are complex conjugates, \( \frac{3}{2}i \) and \( -\frac{3}{2}i \), they will cancel out their imaginary parts when added.

Step 2: Calculate the sum of the roots

The sum of the roots \( \frac{3}{2}i + (-\frac{3}{2}i) \) equals \( 0 \). Therefore, the coefficient of \( x \) in the quadratic equation will be \( 0 \).

Step 3: Calculate the product of the roots

The product of the roots \( (\frac{3}{2}i)(-\frac{3}{2}i) = -\left(\frac{3}{2}\right)^2 = -\frac{9}{4} \).

Step 4: Formulate the quadratic equation

Now we use the format \( x^2 - (p+q)x + pq = 0 \). Since \( p+q = 0 \) and \( pq = -\frac{9}{4} \), the equation becomes \( x^2 + \frac{9}{4} = 0 \). Multiply through by 4 to clear the fraction: \( 4x^2 + 9 = 0 \).

Step 5: Verify the equation

Check the equation \( 4x^2 + 9 = 0 \) by confirming that it gives correct roots. Solving \( 4x^2 + 9 = 0 \) yields \( x^2 = -\frac{9}{4} \), thus \( x = \pm \frac{3}{2}i \), confirming our original roots \( \frac{3}{2}i \) and \(-\frac{3}{2}i \).

Key Concepts

Complex Roots

Complex roots arise when solving quadratic equations where the discriminant (the part under the square root in the quadratic formula) is less than zero. For such equations, the solutions are not real numbers but include imaginary parts. These roots always appear in conjugate pairs.
For instance, if one root is \( \frac{3}{2}i \), its pair will be \( -\frac{3}{2}i \). The imaginary unit \( i \) represents the square root of -1.

• Real roots occur when the discriminant is zero or positive.
• Complex roots show up when the discriminant is negative.

To illustrate, consider the quadratic equation \( 4x^2 + 9 = 0 \). Solving this, we see: \[ x^2 = -\frac{9}{4} \Rightarrow x^2 = -1 \left( \frac{3}{2} \right)^2 \] hence, \( x = \pm \frac{3}{2}i \). Thus, complex roots help in understanding the solutions to equations that cannot be simplified to real numbers alone.

Integer Coefficients

Quadratic equations can have integer coefficients, which makes them easier to handle in both practical and theoretical contexts. When dealing with roots that include fractions or complex numbers, like \( \frac{3}{2}i \) and \( -\frac{3}{2}i \), it's possible to manipulate the equation so that all coefficients become integers.
In our example, the original expression \( x^2 + \frac{9}{4} = 0 \) was modified by multiplying through by 4, resulting in \( 4x^2 + 9 = 0 \).

• Converting coefficients to integers eliminates fractions.
• This process retains the equation's integrity while simplifying calculations.

It's crucial to adjust the entire equation uniformly to maintain balance. Integer coefficients ensure that the equation still accurately represents the relationship between \( x \) and the roots, providing a clean and manageable form.

Complex Conjugates

Complex conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts, such as \( \frac{3}{2}i \) and \( -\frac{3}{2}i \). They are significant in quadratic equations because, together, they ensure coefficients remain real. Why is this important? Because purely real coefficients make equations easier to solve analytically and applicable in real-world scenarios.
In a quadratic equation, like \( 4x^2+9=0 \), complex conjugate roots ( \( \pm \frac{3}{2}i \) ) work seamlessly to cancel out imaginary components in both the sum and product:

• The sum \( \frac{3}{2}i + (-\frac{3}{2}i) = 0 \) results in a real number.
• The product is real as it becomes \( -\left(\frac{3}{2}\right)^2 = -\frac{9}{4} \).

By ensuring the quadratic equation is written with integer coefficients and real components, we create a solid foundation for solving these equations comprehensively.