Problem 42 - College Algebra

Question

Solve the quadratic equation by using the quadratic formula. If the solutions are not real, state No Real Solution. $$ x^{2}+4 x+2=0 $$

Step-by-Step Solution

Answer

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

Step 1: Identify coefficients

The given quadratic equation is \(x^2 + 4x + 2 = 0\). To use the quadratic formula, first identify the coefficients: \(a = 1\), \(b = 4\), and \(c = 2\).

Step 2: Write down the quadratic formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]. We will use this to find the solutions for \(x\).

Step 3: Calculate the discriminant

The discriminant of a quadratic equation is \(b^2 - 4ac\). For the equation, we have: \(b^2 - 4ac = 4^2 - 4 \times 1 \times 2 = 16 - 8 = 8\). This is positive, indicating two distinct real solutions.

Step 4: Substitute into quadratic formula

Substitute the values of \(a\), \(b\), and \(c\) along with the discriminant into the quadratic formula: \[ x = \frac{-4 \pm \sqrt{8}}{2 \times 1} \].

Step 5: Simplify the expression

\(\sqrt{8}\) simplifies to \(2\sqrt{2}\). The solution becomes: \[ x = \frac{-4 \pm 2\sqrt{2}}{2} \]. Simplifying further, we have two solutions: \(x = -2 + \sqrt{2}\) and \(x = -2 - \sqrt{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool in algebra that allows you to find the roots of any quadratic equation, which has the general form, \(ax^2 + bx + c = 0\). It is especially useful when factoring is not obvious or easy. The key formula to remember is:
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
In this elegant equation:

• \(a\), \(b\), and \(c\) are the coefficients of the quadratic equation where \(a\) is the coefficient of \(x^2\), \(b\) is the coefficient of \(x\), and \(c\) is the constant term.
• The "±" symbol means we will perform two separate calculations: one using the "+" and another with the "−".

To use the quadratic formula, simply plug the corresponding values of \(a\), \(b\), and \(c\) into the formula. Be careful with arithmetic, and remember to simplify when possible. This formula will work as long as \(aeq 0\), as a zero value would make the equation not quadratic. The quadratic formula is a powerful method because it guarantees a solution whenever one exists.

Discriminant

The discriminant is a specific part of the quadratic formula and plays a crucial role in determining the nature of the roots of the quadratic equation. It is found under the square root sign in the quadratic formula, given by:
\[ D = b^2 - 4ac \]
The value of the discriminant (D) reveals whether the solutions are real and how many there are:

• If \(D > 0\), there are two distinct real solutions. This situation means the parabola intersects the x-axis at two points.
• If \(D = 0\), there is exactly one real solution, also known as a repeated or double root. Here, the parabola just touches the x-axis.
• If \(D < 0\), there are no real solutions. In this case, the solutions are complex or imaginary numbers, and the graph does not touch the x-axis at all.

Understanding the discriminant is helpful because it quickly tells you what to expect from the solutions even before performing full calculations. It saves time when the problem asks whether any real solutions exist for a quadratic equation.

Real Solutions

Real solutions of a quadratic equation refer to the points where the graph of the equation intersects the x-axis. These solutions are the x-values that satisfy the equation \(ax^2 + bx + c = 0\) with real numbers. Here's what you need to know:

• These solutions can either be two distinct values, one unique value, or none at all, depending on the discriminant’s value.
• For instance, when the discriminant \(D > 0\), the quadratic equation has two distinct real solutions. This is typical when the quadratic equation's graph looks like a "U" or an inverted "U" intersecting the x-axis at two points.
• If \(D = 0\), the graph touches the x-axis at one point, producing a repeated real solution.
• However, when \(D < 0\), the graph does not intersect the x-axis because there are no real solutions.

Recognizing the nature of the solutions can help visualize the problem and confirms that computations using the quadratic formula are on track. Real solutions represent tangible, measurable outcomes of the quadratic function's behavior when plotted on a graph.