Problem 43 - Algebra 2

Question

Solve each equation using the Quadratic Formula. $$ 2\left(x^{2}+2\right)=3 x $$

Step-by-Step Solution

Answer

The solutions to the equation \(2(x^{2}+2)=3x\) are \(x = \frac{3}{4} \pm \frac{\sqrt{23}}{4}i\).

Step 1: Simplify the Equation

First, distribute the 2 on the left side of the equation and then move all terms to one side to get the equation in standard quadratic form, \(ax^2 + bx + c = 0\). The equation becomes \(2x^2 - 3x + 4 = 0\).

Step 2: Identify Coefficients

Identify the coefficients of the quadratic equation \(ax^2 + bx + c = 0\), where \(a=2\), \(b=-3\), and \(c=4\).

Step 3: Apply the Quadratic Formula

Insert the coefficients into the Quadratic Formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\). This gives us \(x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(2)(4)}}{2(2)}\).

Step 4: Simplify Under the Square Root

Calculate the discriminant \(\Delta = b^2 - 4ac\) which is \((-3)^2 - 4(2)(4) = 9 - 32 = -23\).

Step 5: Calculate the Solutions

Since the discriminant is negative, the quadratic equation has two complex solutions: \(x = \frac{3 \pm \sqrt{-23}}{4}\).

Step 6: Express the Solutions

Write the final solutions by expressing the square root of a negative number as \(i\times\sqrt{23}\). The solutions are \(x = \frac{3}{4} \pm \frac{\sqrt{23}}{4}i\).

Key Concepts

Quadratic Equation

The quadratic equation is a second-order polynomial equation of the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are coefficients, \(a \eq 0\), and \(x\) represents an unknown variable. It's referred to as 'quadratic' because 'quadra' means 'square' in Latin, reflecting the \(x^2\) term. Quadratic equations are fundamental in algebra and are used to model various real-world situations, such as calculating areas, determining product pricing, and predicting profits.

For example, the quadratic equation in the original exercise, \(2x^2 - 3x + 4 = 0\), requires us to find the value of \(x\) that makes the equation true. Understanding how to manipulate and solve these equations is essential for students, as it forms the basis for more advanced mathematical concepts.

Solving Quadratic Equations

Solving quadratic equations can be done in several ways including factoring, completing the square, graphing, and using the Quadratic Formula. The most universal method, applicable to all forms of quadratic equations, is the Quadratic Formula: \[\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\].

Using this formula involves identifying the coefficients \(a\), \(b\), and \(c\) from the standard form of the quadratic equation and then substituting these values into the formula to find the solutions for \(x\). It is a foolproof method that provides solutions in a systematic manner but requires a good understanding of algebraic manipulation and dealing with square roots.

Discriminant

The discriminant is a component of the Quadratic Formula and is specifically the value under the square root symbol, denoted by \(\Delta \): \(\Delta = b^2 - 4ac \).

It plays a crucial role in determining the nature of the roots of a quadratic equation:

• If \(\Delta > 0\), the equation has two distinct real solutions.
• If \(\Delta = 0\), the equation has exactly one real solution (also called a repeated root).
• If \(\Delta < 0\), the equation has two complex solutions.

In the provided exercise, the discriminant calculation shows a negative result \((-23\)), indicating that our equation will have complex solutions as opposed to real ones.

Complex Numbers

Complex numbers extend the idea of the number line to a two-dimensional plane by introducing an imaginary unit \(i\). A complex number is expressed as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. When dealing with quadratic equations whose discriminants are negative, like in our exercise, complex numbers come into play.

The term \(i\) is defined as \(i^2 = -1\). Solving our equation leads us to use the imaginary unit because we end up taking the square root of a negative number. Therefore, the final solutions include \(i\), like \(\frac{3}{4} \pm \frac{\sqrt{23}}{4}i\), signifying that we are dealing with numbers that cannot be found on the traditional number line.