Problem 42 - College Algebra with Corequisite Support

Question

For the following exercises, 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 the coefficients

For a quadratic equation in the standard form \(ax^2 + bx + c = 0\), identify the coefficients \(a\), \(b\), and \(c\). In the equation \(x^2 + 4x + 2 = 0\), we have \(a = 1\), \(b = 4\), and \(c = 2\).

Step 2: Write down the quadratic formula

The quadratic formula is used to find the solutions of a quadratic equation and is given by \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).

Step 3: Calculate the discriminant

The discriminant \(D\) is a part of the quadratic formula that is under the square root, given by \(b^2 - 4ac\). Calculate it as follows: \(D = 4^2 - 4 \times 1 \times 2 = 16 - 8 = 8\). The discriminant \(D = 8\) is greater than zero, which means the equation has two real solutions.

Step 4: Substitute the values into the quadratic formula

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

Step 5: Simplify the expression under the square root

The square root of 8 can be simplified to \(2\sqrt{2}\). Substitute to get \(x = \frac{-4 \pm 2\sqrt{2}}{2}\).

Step 6: Simplify the fraction

Divide each term in the numerator by 2: \(x = \frac{-4}{2} \pm \frac{2\sqrt{2}}{2}\). Simplifying gives \(x = -2 \pm \sqrt{2}\). Thus, the solutions are \(x = -2 + \sqrt{2}\) and \(x = -2 - \sqrt{2}\).

Key Concepts

Quadratic Formula

The quadratic formula is a fundamental tool for solving quadratic equations, which are polynomial equations of degree 2. It provides an efficient way to find the values of x that satisfy the equation.
For a generic quadratic equation written in the form \( ax^2 + bx + c = 0 \), the quadratic formula is expressed as:
\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]Here, \( a \), \( b \), and \( c \) are coefficients of the terms in the quadratic equation, and \( x \) represents the variable whose values we want to determine.
The symbol "±" indicates there can be two possible solutions because of the square root function. This formula is essential as it provides a solution irrespective of whether the roots are real or complex.

Discriminant

The discriminant is a critical component in the quadratic formula and is a determinant factor for the nature of the roots of a quadratic equation. It is the part under the square root:
\[D = b^2 - 4ac\]The value of the discriminant determines the number and type of solutions:

• If \( D > 0 \), the quadratic equation has two distinct real solutions.
• If \( D = 0 \), there is exactly one real solution (or two identical real solutions).
• If \( D < 0 \), there are no real solutions as the solutions are complex numbers.

In our case, with a discriminant of \( D = 8 \), which is greater than zero, it indicates the equation has two separate real solutions.

Real Solutions

In the context of a quadratic equation, real solutions are the x-values that satisfy the equation and are represented by real numbers.
Real solutions occur when the discriminant \( D \) is greater than or equal to zero. Each real solution corresponds to a point where the parabola, defined by the quadratic equation, intersects the x-axis.
The solutions can be found using the quadratic formula. For the example \( x^2 + 4x + 2 = 0 \), the solutions are calculated as:
\( x = -2 + \sqrt{2} \) and \( x = -2 - \sqrt{2} \). These solutions are real because they were obtained using a positive discriminant.

Simplifying Expressions

Simplifying expressions is an important step in solving quadratic equations through the quadratic formula. It involves reducing complex expressions into their simplest form to obtain clear and manageable solutions.
This often includes simplifying the square root and the fractions:

• The square root of 8 simplifies to \( 2\sqrt{2} \).
• The expression \( \frac{-4 \pm 2\sqrt{2}}{2} \) simplifies to \( -2 \pm \sqrt{2} \).

By simplifying, we make the solutions more understandable and concise. This process of simplification ensures the final answers are clear and are in their most reduced forms, making them easier to interpret and verify.