Problem 27 - Algebra 2

Question

Solve each equation by using the method of your choice. Find exact solutions. \(x^{2}-4 x+7=0\)

Step-by-Step Solution

Answer

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

Step 1: Identify the Equation Type

The equation given is a quadratic equation with the form \( ax^{2} + bx + c = 0 \), where \(a = 1\), \(b = -4\), and \(c = 7\). It doesn't factor nicely, so we'll use the quadratic formula.

Step 2: Apply the Quadratic Formula

The quadratic formula is \( x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \). Here, \(a = 1\), \(b = -4\), and \(c = 7\). Plug these values into the formula.

Step 3: Calculate the Discriminant

The discriminant is given by \( b^{2} - 4ac \). Substituting the values, we get \((-4)^{2} - 4 \times 1 \times 7 = 16 - 28 = -12\). Since the discriminant is negative, the solutions are complex.

Step 4: Solve Using the Quadratic Formula

Substitute the discriminant back into the quadratic formula: \( x = \frac{4 \pm \sqrt{-12}}{2} \). Simplify to get \( x = \frac{4 \pm \sqrt{4 \times 3}i}{2} = \frac{4 \pm 2i \sqrt{3}}{2} \).

Step 5: Simplify the Solutions

Simplify \( \frac{4 \pm 2i \sqrt{3}}{2} \) to get \( x = 2 \pm i \sqrt{3} \). Thus, the solutions are \( x = 2 + i \sqrt{3} \) and \( x = 2 - i \sqrt{3} \).

Key Concepts

Quadratic Formula

The quadratic formula is a powerful tool for finding the solutions of a quadratic equation. A quadratic equation has the standard form \( ax^2 + bx + c = 0 \). This formula is:

• \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)

Here, \( a \), \( b \), and \( c \) are coefficients of the equation. This formula is indispensable when the quadratic doesn't factor easily.
It tells us how to find the exact values of \( x \) that make the equation true, even when the solutions are not nice whole numbers.
Substituting the values of \( a \), \( b \), and \( c \) directly into the formula gives us the solutions in one step.

Discriminant

The discriminant plays a crucial role in determining the nature of the solutions of a quadratic equation. It is the part under the square root in the quadratic formula, given by:

• \( b^2 - 4ac \)

The value of the discriminant can tell us:

• If it is positive, there are two distinct real solutions.
• If it is zero, there is exactly one real solution.
• If it is negative, as in this exercise with \(-12\), it indicates that the solutions are complex numbers.

Understanding the discriminant can give us a quick sense of what kinds of solutions to expect, saving time when solving the equation.

Complex Solutions

Complex solutions arise when the discriminant of a quadratic equation is negative. These solutions involve the imaginary unit \( i \), where \( i = \sqrt{-1} \). In our example, the discriminant was \(-12\), leading to solutions that include imaginary numbers.The quadratic formula will yield complex solutions in the form of:

• \( x = p \pm qi \)

Where \( p \) and \( q \) are real numbers.
For the equation \( x^2 - 4x + 7 = 0 \), the solutions were calculated as \( x = 2 \pm i\sqrt{3} \). These solutions mean that they do not lie on the real number line, but instead in the complex plane.

Solving Equations

Solving quadratic equations involves finding the values of \( x \) that satisfy the equation. The method we used here was the quadratic formula, which is especially useful when equations cannot be easily factored.To solve:

• Identify \( a \), \( b \), and \( c \) from \( ax^2 + bx + c = 0 \).
• Compute the discriminant \( b^2 - 4ac \).
• Insert these into the quadratic formula.
• Solve the resulting equation, simplifying as necessary to obtain the solutions.

Using these steps ensures you obtain the right solutions, highlighting the importance of structured problem-solving.

Polynomial Equations

Polynomial equations such as quadratics are those that involve sums of powers of variables. A quadratic is a second-degree polynomial, meaning the highest exponent is 2. These types of equations can model real-world phenomena and provide foundational understanding for more complex mathematics.
Recognizing the form of polynomial equations helps in deciding which method to use for solving:

• Factoring (when convenient and possible)
• Applying the quadratic formula
• Utilizing graphing techniques or numerical methods

Being familiar with the polynomial nature of these equations is essential for recognizing solutions that may involve both real and complex numbers.