Problem 99

Question

Solve equation by the method of your choice. $$ x^{2}-6 x+13=0 $$

Step-by-Step Solution

Verified

Answer

The solutions to the equation are $x_{1} = 3 + 2i$ and $x_{2} = 3 - 2i$. These are complex roots, which occur because the discriminant of the equation is less than zero.

1Step 1: Identifying coefficients

From comparing the given equation $x^{2}-6 x+13=0$ with the standard quadratic equation $ax^{2} + bx + c = 0$, we get coefficients as follows: $a = 1$, $b = -6$, and $c = 13$.

2Step 2: Calculate the discriminant

The discriminant is given by $b^{2} - 4ac$. Substituting the values of $a$, $b$, and $c$, we get $(-6)^{2} - 4*1*13 = 36 - 52 = -16$. Since the discriminant is less than zero, the roots of the equation will be complex or imaginary numbers.

3Step 3: Use the Quadratic Formula

Next, we find the roots $x_{1}$ and $x_{2}$ of the equation using the quadratic formula: $x_{i} = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$. Substituting values of $a$, $b$, and $c$ we get: $x_{1} = \frac{6 + \sqrt{-16}}{2} = 3 + 2i$ and $x_{2} = \frac{6 - \sqrt{-16}}{2} = 3 - 2i$.

Key Concepts

Quadratic FormulaDiscriminant of a Quadratic EquationComplex Roots

Quadratic Formula

When it comes to solving quadratic equations, the quadratic formula stands out as a powerful tool that can find the roots of any quadratic equation. A quadratic equation is typically in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are coefficients, and $x$ represents the unknown variable.

The quadratic formula is expressed as:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]. This formula calculates the solutions by considering the coefficients of the equation. To use the formula, one simply substitutes the values of $a$, $b$, and $c$ into the formula, performing the calculations inside the square root (known as the discriminant), and dividing by $2a$.

Importance of the $\pm$ Sign

The $\pm$ sign indicates two possible values for $x$: one resulting from adding the square root of the discriminant, and another from subtracting it. This corresponds to the two potential roots of the quadratic equation. As such, the quadratic formula provides a systematic and effortless approach for solving any quadratic equation, whether it leads to real or complex roots.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation, symbolized as $\Delta$, is a key component that determines the nature of the roots the equation will yield. It is calculated using the formula $\Delta = b^2 - 4ac$, based on the coefficients of the standard quadratic equation $ax^2 + bx + c = 0$.

Three scenarios can arise from the discriminant's value:

If $\Delta > 0$, there are two distinct real roots.
If $\Delta = 0$, there is exactly one real root (also called a repeated or double root).
If $\Delta < 0$, the roots are complex and come in a conjugate pair, meaning they have the form $a \pm bi$ where $i$ is the imaginary unit and represents the square root of $-1$.

Calculating the Discriminant

For the equation $x^2 - 6x + 13 = 0$, the discriminant calculation would be $(-6)^2 - 4(1)(13)$, resulting in $-16$. This negative value instantly informs us that the equation's roots will not be real numbers but complex ones.

Complex Roots

Complex roots occur when the discriminant of a quadratic equation is negative. Given the standard quadratic equation $ax^2 + bx + c = 0$, a negative discriminant implies that there are no real solutions for $x$ that can satisfy the equation. Instead, the roots will be complex numbers, which contain both a real part and an imaginary part.

In mathematics, the imaginary unit $i$ represents $\sqrt{-1}$, which opens the door to complex numbers. A complex number looks like $a + bi$, where $a$ is the real part, and $bi$ is the imaginary part.

Identifying Complex Roots

For instance, considering the equation from the exercise, $x^2 - 6x + 13 = 0$, after applying the quadratic formula, we find that the roots are $3 + 2i$ and $3 - 2i$. The $3$ represents the real part that is the same for both roots, while $\pm2i$ indicates the imaginary parts. These complex roots always come in conjugate pairs. Understanding complex roots is essential for anyone delving into advanced mathematics, as they are widely used in fields such as engineering, physics, and applied mathematics.

Problem 99