Problem 80 - College Algebra

Question

This will help you prepare for the material covered in the next section. $$ \text { Solve: } x^{2}+4 x+6=0 $$

Step-by-Step Solution

Answer

The solutions to the equation are \(-2 + i\sqrt{2}\) and \(-2 - i\sqrt{2}\).

Step 1: Identify a, b, and c

From the quadratic equation \(x^{2}+4x+6=0\), we can see that the coefficients are a = 1, b = 4, and c = 6.

Step 2: Calculate the discriminant \(b^2-4ac\)

The discriminant can be calculated as follows: \(b^2-4ac = 4^{2} - 4(1)(6) = 16 - 24 = -8\).

Step 3: Apply the Quadratic formula

The solutions to the equation are given by \(\frac{-b \pm \sqrt{b^2-4ac}}{2a}\). Substituting the values for a, b, c and the discriminant, we get: \(\frac{-4 \pm \sqrt{-8}}{2(1)} = \frac{-4 \pm 2i\sqrt{2}}{2} = -2 \pm i\sqrt{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool used to find the solutions, or zeros, of a quadratic equation. A quadratic equation is given in the form of \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(a\) is not zero.

The quadratic formula itself is \(x = \frac{-b \[+\] \[ b^2-4ac \]}{2a}\), with \(\[ \]\b2-4ac\) representing the discriminant. To use it effectively:

• First, identify the coefficients \(a\), \(b\), and \(c\) from your quadratic equation.
• Then, insert these values into the formula, performing the arithmetic carefully to find the values of \(x\).

In our example, \(x^2+4x+6=0\), we have \(a=1\), \(b=4\), and \(c=6\). These are plugged into the formula as shown in the provided step by step solution, yielding the roots of the equation.

Discriminant of a Quadratic Equation

The discriminant of a quadratic equation is a part of the quadratic formula enclosed in the square root, symbolized as \(b^2-4ac\). It is a powerful indicator as it determines the nature and number of solutions.

• If the discriminant is positive, the equation has two distinct real solutions;
• If it is zero, there is one real solution, a 'perfect square';
• If negative, the equation has two complex solutions.

In the given example, the discriminant is \(-8\), which is less than zero. This informs us immediately that the quadratic equation does not have real roots. Instead, it has two complex roots, as the step by step solution correctly identifies. Remember, it's the sign of the discriminant, rather than its magnitude, that tells us about the types of solutions.

Complex Numbers in Quadratics

When the discriminant of a quadratic equation is negative, the solutions involve complex numbers. Complex numbers are made up of a real and an imaginary part, usually represented as \(a + bi\), where \(i\) is the imaginary unit (\(i^2 = -1\)).

In the context of quadratic equations with no real roots, the quadratic formula will yield answers in the form of complex conjugates, \(-p \[+\] qi\), where \(p\) and \(q\) are real numbers and \(q\) is the positive square root of the negative discriminant.

• This demonstrates that even equations that can't be solved by finding points where the parabola crosses the x-axis (because it doesn't cross it) still have valid solutions in the complex plane.

The final answers in our example equation, \(-2 \[+\] i\b2\), are indeed two complex numbers, showcasing the full application of the quadratic formula in the presence of a negative discriminant.