Problem 59 - Elementary Algebra

Question

Solve using the quadratic formula. $$ x 2+4 x+6=0 $$

Step-by-Step Solution

Answer

The solutions are \(x = -2 \pm \sqrt{2}i\).

Step 1: Identify the coefficients

The quadratic equation is given as \(x^2 + 4x + 6 = 0\). Identify the coefficients: \(a = 1\), \(b = 4\), and \(c = 6\). These will be used in the quadratic formula.

Step 2: Write down the quadratic formula

The quadratic formula is used to find the solutions of the quadratic equation \(ax^2 + bx + c = 0\) and is given by:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]

Step 3: Calculate the discriminant

The discriminant \(D\) is calculated as \(b^2 - 4ac\). Substitute the values: \(b = 4\), \(a = 1\), \(c = 6\) to get:\[ D = 4^2 - 4 \cdot 1 \cdot 6 = 16 - 24 = -8 \]

Step 4: Interpret the discriminant

Since the discriminant \(D = -8\) is negative, this means the quadratic equation has two complex (imaginary) solutions.

Step 5: Substitute into the quadratic formula

Since the solutions are complex, substitute into the quadratic formula:\[ x = \frac{-4 \pm \sqrt{-8}}{2 \times 1} \]Calculate the square root of \(-8\): \(\sqrt{-8} = \sqrt{8}i = 2\sqrt{2}i\).

Step 6: Simplify the expression

Continue by simplifying the expression:\[ x = \frac{-4 \pm 2\sqrt{2}i}{2} \]Break it down for clarity:\[ x = \frac{-4}{2} \pm \frac{2\sqrt{2}i}{2} \] This simplifies to:\[ x = -2 \pm \sqrt{2}i \]

Key Concepts

Discriminant

The discriminant is a crucial part of the quadratic formula and plays a key role in determining the nature of the roots of a quadratic equation. It is calculated as part of the expression under the square root in the quadratic formula:

• Quadratic Formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
• Discriminant: \( D = b^2 - 4ac \)

The value of the discriminant tells you what kind of solutions to expect. Here's what the different values mean:

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

In our problem \( x^2 + 4x + 6 = 0 \), we calculated the discriminant to be \( -8 \). This negative value indicates that the solutions are complex numbers.

Complex Solutions

When solving quadratic equations with a negative discriminant, the solutions are complex numbers. Complex numbers have both a real part and an imaginary part. The imaginary unit is represented by \( i \), where \( i^2 = -1 \).For our example, the quadratic equation \( x^2 + 4x + 6 = 0 \) has a discriminant of \(-8\). Using the quadratic formula, the complex solutions can be found as follows:

• First, note that \( \sqrt{-8} \) becomes \( \sqrt{8}i \), which simplifies to \( 2\sqrt{2}i \).
• Substitute into the quadratic formula: \( x = \frac{-4 \pm 2\sqrt{2}i}{2} \).
• Simplify to get the solutions: \( x = -2 \pm \sqrt{2}i \).

Thus, the two complex solutions are rooted in a real part \(-2\) and an imaginary part \( \pm \sqrt{2}i \). These solutions mean the parabola does not intersect the x-axis at any real point.

Solving Quadratic Equations

To solve quadratic equations like \( ax^2 + bx + c = 0 \), the quadratic formula is a universally applicable tool:

• Identify the coefficients \( a \), \( b \), and \( c \) from the equation.
• Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to find the solutions.

The process involves:

• Identifying \( a = 1 \), \( b = 4 \), and \( c = 6 \) in our problem.
• Calculating the discriminant \( D \).
• Substituting the values into the formula.
• Simplifying to find the solutions. For complex solutions, remember \( i = \sqrt{-1} \).

This method ensures you identify the nature of the solutions (real or complex) and accurately find them. It's a powerful technique because it works for any quadratic equation, regardless of the coefficients or discriminant value.