Problem 57 - Algebra 2

Question

PREREQUISITE SKILL. Find the exact solutions of each equation by using the Quadratic Formula. $$ 2 x^{2}+3 x+2=0 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{-3 \pm i\sqrt{7}}{4} \).

Step 1: Identify Coefficients

The quadratic equation is given in the form \( ax^2 + bx + c = 0 \). For the equation \( 2x^2 + 3x + 2 = 0 \), we identify the coefficients as: \( a = 2 \), \( b = 3 \), and \( c = 2 \).

Step 2: Write the Quadratic Formula

The quadratic formula is used to find the solutions of a quadratic equation, and it is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \].

Step 3: Calculate the Discriminant

Calculate the discriminant using \( b^2 - 4ac \). Substitute the values: \( b = 3 \), \( a = 2 \), and \( c = 2 \). So, \( b^2 - 4ac = 3^2 - 4 \times 2 \times 2 = 9 - 16 = -7 \).

Step 4: Analyze the Discriminant

Since the discriminant \(-7\) is less than zero, this means there are two complex solutions to the equation.

Step 5: Apply the Quadratic Formula

Since the solutions are complex, apply the quadratic formula: \[ x = \frac{-3 \pm \sqrt{-7}}{4} \]. This can be rewritten with the imaginary unit: \[ x = \frac{-3 \pm i\sqrt{7}}{4} \].

Step 6: Write the Complex Solutions

The two solutions of the quadratic equation are: \( x = \frac{-3 + i\sqrt{7}}{4} \) and \( x = \frac{-3 - i\sqrt{7}}{4} \).

Key Concepts

Complex Solutions

In mathematics, complex solutions arise when solving quadratic equations with a negative discriminant. A complex solution means that the result of the quadratic formula includes imaginary numbers. - Complex numbers have a real part and an imaginary part. An imaginary number is defined using the imaginary unit, denoted as \(i\). The value of \(i\) is such that \(i^2 = -1\).- When the discriminant \(b^2 - 4ac\) is less than zero, it indicates that the square root of a negative number is involved.
- This situation produces solutions with imaginary numbers, often expressed in the form \(a + bi\), where \(a\) and \(b\) are real numbers.
In our example, the quadratic equation \(2x^2 + 3x + 2 = 0\) yields complex solutions \( x = \frac{-3 \pm i\sqrt{7}}{4} \). Here, \(i\sqrt{7}\) represents the imaginary part of the solution.

Discriminant

The discriminant is a key part of the quadratic formula and is represented by the expression \(b^2 - 4ac\). It helps determine the nature of the roots of a quadratic equation:- If the discriminant is positive, the quadratic equation has two distinct real roots.
- If it is zero, the equation has exactly one real root, sometimes referred to as a repeated or double root.
- When the discriminant is negative, as in \(9 - 16 = -7\), it indicates that the roots of the equation are complex rather than real. This tells us that the solutions involve imaginary numbers. The presence of complex solutions often involves mathematical elegance in expressing them, as they include both real and imaginary components.
Understanding the discriminant's role is crucial as it quickly reveals a lot about the equations you're handling, without necessarily solving them right away.

Quadratic Equation

A quadratic equation is a polynomial equation of the second degree, typically written as \(ax^2 + bx + c = 0\). In this representation:- \(a\), \(b\), and \(c\) are constants with \(a eq 0\). The term \(a\) multiplying \(x^2\) guarantees that the equation is quadratic.
- Such equations are graphically represented as parabolas when plotted on a coordinate plane.
To solve a quadratic equation, you can use different methods such as factoring, completing the square, or employing the quadratic formula. The quadratic formula, \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), is a powerful universal tool that can find solutions for any quadratic equation, including those that can't be factored neatly.
For the equation \(2x^2 + 3x + 2 = 0\), applying the quadratic formula reveals important insights about the nature of its solutions. This equation results in complex solutions due to its negative discriminant.