Question

Construct the quadratic equations that have the following pairs of roots: (a) \(-6,-3 ;\) (b) 0,\(4 ;\) (c) 2,\(2 ;\) (d) \(3+2 i, 3-2 i\), where \(i^{2}=-1 .\)

Short answer

(a) \(x^2 + 9x + 18 = 0\), (b) \(x^2 - 4x = 0\), (c) \(x^2 - 4x + 4 = 0\), (d) \(x^2 - 6x + 13 = 0\).

Step-by-step solution

- General Form of Quadratic Equation

Recall the general form of a quadratic equation: \[ ax^2 + bx + c = 0 \] If the roots of the equation are \( p \) and \( q \), then the quadratic equation can be constructed as: \[ (x - p)(x - q) = 0 \].

- Construct Equation for Roots \(-6, -3\)

For roots \( -6 \) and \( -3 \): \[ (x - (-6))(x - (-3)) = 0 \] \[ (x + 6)(x + 3) = 0 \]. Expanding this, we get: \[ x^2 + 3x + 6x + 18 = 0 \] \[ x^2 + 9x + 18 = 0 \].

- Construct Equation for Roots \(0, 4\)

For roots \( 0 \) and \( 4 \): \[ (x - 0)(x - 4) = 0 \] \[ x(x - 4) = 0 \]. Expanding this, we get: \[ x^2 - 4x = 0 \].

- Construct Equation for Roots \(2, 2\)

For roots \( 2 \) and \( 2 \): \[ (x - 2)(x - 2) = 0 \] \[ (x - 2)^2 = 0 \]. Expanding this, we get: \[ x^2 - 4x + 4 = 0 \].

- Construct Equation for Roots \(3 + 2i, 3 - 2i\)

For roots \( 3 + 2i \) and \( 3 - 2i \): \[ (x - (3 + 2i))(x - (3 - 2i)) = 0 \] Using the difference of squares formula, \[ (x - 3 - 2i)(x - 3 + 2i) = 0 \] \[ (x - 3)^2 - (2i)^2 = 0 \] \[ x^2 - 6x + 9 - 4(-1) = 0 \] \[ x^2 - 6x + 13 = 0 \].

Key concepts

roots of quadratic equations

The roots of a quadratic equation are the values of \(x\) that satisfy the equation \(ax^2 + bx + c = 0\). These roots can be found by factoring the equation, using the quadratic formula, or completing the square.
For a quadratic equation with roots \(p\) and \(q\), we can express it as: \[ (x - p)(x - q) = 0 \]
Expanding this product will yield the standard quadratic form.
For example, if the roots are \(-6\) and \(-3\), then the corresponding quadratic equation is: \[ (x + 6)(x + 3) = 0 \] Expanding gives \[ x^2 + 9x + 18 = 0 \].
Similarly, different pairs of roots lead to different quadratic equations, as shown in the exercise.

complex numbers

Complex numbers are numbers in the form \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with the property \(i^2 = -1\).
In quadratic equations, complex roots often appear in conjugate pairs, such as \(3 + 2i\) and \(3 - 2i\).
To construct a quadratic equation from these roots, we use the same approach but apply the difference of squares formula: \[ (x - (3 + 2i))(x - (3 - 2i)) = 0 \] Which simplifies to: \[ (x - 3)^2 - (2i)^2 = 0 \] Given that \(i^2 = -1\), we have: \[ x^2 - 6x + 9 + 4 = 0 \] resulting in the quadratic equation: \[ x^2 - 6x + 13 = 0 \].
This kind of equation illustrates the relationship between complex conjugates and real coefficients in quadratic polynomials.

polynomial expansion

Polynomial expansion involves expressing products of polynomials as a sum of monomials.
For quadratic equations, it means multiplying binomials to get the standard quadratic form.
If you have the roots \(p\) and \(q\), the equation \[ (x - p)(x - q) = 0 \] expands by the distributive property:
For roots \(-6\) and \(-3\): \[ (x + 6)(x + 3) = x(x + 3) + 6(x + 3) = x^2 + 3x + 6x + 18 = x^2 + 9x + 18 = 0 \].
Each term arises from combining each term in the first binomial with every term in the second binomial.
This process ensures we return the equation to its standard form: \[ ax^2 + bx + c = 0 \].

difference of squares

The difference of squares is a useful algebraic formula: \[ a^2 - b^2 = (a - b)(a + b) \].
This helps simplify products involving complex numbers.
For example, given roots \(3 + 2i\) and \(3 - 2i\), we use the formula: \[ (x - (3 + 2i))(x - (3 - 2i)) = (x - 3)^2 - (2i)^2 = 0 \].
Applying \(i^2 = -1\), simplifies to: \[ x^2 - 6x + 9 - 4(-1) = x^2 - 6x + 13 = 0 \].
This transformation shows how the formula turns a seemingly complicated product into a manageable quadratic form.