Question

Solve each equation in the complex number system. $$ x^{2}-6 x+13=0 $$

Short answer

The solutions are \( x = 3 + 2i \) and \( x = 3 - 2i \).

Step-by-step solution

- Write down the quadratic equation

The given quadratic equation is \[ x^2 - 6x + 13 = 0 \]

- Identify the coefficients

In the quadratic equation \[ ax^2 + bx + c = 0 \]the coefficients are:a = 1,b = -6,c = 13

- Calculate the discriminant

The discriminant (Δ) is calculated using the formula:\[ Δ = b^2 - 4ac \] Substitute the values of a, b, and c: \[ Δ = (-6)^2 - 4(1)(13) = 36 - 52 = -16 \] Since Δ is negative, the equation has complex roots.

- Use the quadratic formula

The roots of the quadratic equation can be found using the quadratic formula: \[ x = \frac{-b \pm \sqrt{Δ}}{2a} \] Substitute the values of a, b, and Δ: \[ x = \frac{6 \pm \sqrt{-16}}{2} \]

- Simplify the expression

Simplify the expression under the square root and then solve for x: \[ \sqrt{-16} = 4i \] Therefore,\[ x = \frac{6 \pm 4i}{2} \]which simplifies to \[ x = 3 \pm 2i \]

Key concepts

complex roots

When solving quadratic equations, it's possible to encounter complex roots, especially when the discriminant is negative. Complex roots arise from equations that cannot be solved using real numbers alone. They involve the imaginary unit 'i', which represents the square root of -1. In the equation we solved, we obtained complex roots because the discriminant was negative (\textminus16). These roots are in the form of:

• \(x = 3 + 2i\)
• \(x = 3 - 2i\)

Complex roots often come in conjugate pairs, meaning they are mirror images of each other across the real number line. This is evident in our roots 3 + 2i and 3 - 2i.
Understanding complex roots is essential in mathematics because they broaden the scope of solutions, which is particularly important in advanced studies and applications.

quadratic formula

The quadratic formula is a powerful tool for solving quadratic equations of the form: \(ax^2 + bx + c = 0\). The formula is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
By just plugging in the values of a, b, and c, you can find the roots of any quadratic equation. It is particularly useful because it always works, regardless of whether the roots are real or complex.
Let’s use our example:

• a = 1
• b = -6
• c = 13

Substituting these into the quadratic formula, we get:
\( x = \frac{-(-6) \pm \sqrt{(-6)^2 - 4(1)(13)}}{2(1)} \)
Simplifying further:
\(x = \frac{6 \pm \sqrt{36 - 52}}{2} \)
\(x = \frac{6 \pm \sqrt{-16}}{2}\)
Since the discriminant is negative, we recognize the presence of complex roots. This shows how the quadratic formula is straightforward but can lead to complex solutions depending on the discriminant value.

discriminant

The discriminant in a quadratic equation provides crucial information about the nature of the roots. It is calculated using the formula:
\[ Δ = b^2 - 4ac \] The value of the discriminant determines the type of roots you will get:

• If Δ > 0, the equation has two distinct real roots.
• If Δ = 0, the equation has exactly one real root (also known as a repeated or double root).
• If Δ < 0, the equation has two complex conjugate roots.

In our example:

• b = -6
• a = 1
• c = 13

Substituting these into the discriminant formula:
\( Δ = (-6)^2 - 4(1)(13) = 36 - 52 = -16 \)
Here the discriminant Δ is -16, which is less than zero. This negative value indicates that our quadratic equation has complex roots, as we found in the solution. Understanding the discriminant is vital because it lets us predict the necessary method to find the roots and the type of numbers (real or complex) expected.