Problem 95 Solve each equation. Write the a... [FREE SOLUTION]

91影视

Beginning Algebra: A Guided Approach

Rosemary Karr, Marilyn Massey, R. David Gustafson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 9: Problem 95

Solve each equation. Write the answer in bi or a $+$ bi form. $$3 x^{2}-4 x+2=0$$

Short Answer

Expert verified

The solutions to the given equation are $x= \frac{2}{3} \pm \frac{2\sqrt{2}}{3}i$

Step by step solution

Identify a, b, and c

From the equation, the coefficients corresponding to a, b, and c in the quadratic formula are a = 3, b = -4 and c = 2.

Apply the quadratic formula

The quadratic formula is given by $x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}$. Substituting the known values in, we get $x = \frac{4 \pm \sqrt{(-4)^{2} - 4*3*2}}{2*3}$

Simplify the expression under the square root

Simplify the term inside the square root to get $16 - 24 = -8$. Therefore, the equation becomes $x = \frac{4 \pm \sqrt{-8}}{6}$

Handle the negative under the square root

The square root of a negative number introduces the imaginary number i, where $i^{2} = -1$, thus we can rewrite the above equation as $x = \frac{4 \pm \sqrt{8}*i}{6}$

Simplify final results

Further simplifying gives $x= \frac{2}{3} \pm \frac{2\sqrt{2}}{3}i$ which represent the two solutions to the quadratic equation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Quadratic Equation

A quadratic equation is one of the most common types of polynomial equations that you will encounter. It is typically written in the standard form:

$ ax^2 + bx + c = 0 $

where $a$, $b$, and $c$ are constants, and $x$ represents the unknown variable. The coefficient $a$ cannot be zero, as that would make the equation linear rather than quadratic. The highest degree term with $x^2$ defines the equation's classification as quadratic. These equations often have two solutions or "roots" that can be real or complex numbers. Understanding quadratic equations is foundational as they frequently appear in various branches of mathematics and science.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. When you have an equation in the form $ax^2 + bx + c = 0$, you can find its solutions using:

$ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} $

This formula arises from the process of completing the square on the general quadratic equation and is used to calculate the roots directly. The terms within the formula have specific roles:

$-b$ provides the necessary shift caused by the linear term.
$\pm$ indicates that there can be two potential solutions.
The square root $\sqrt{b^2 - 4ac}$ is known as the discriminant and determines the nature of the roots.

If the discriminant is positive, you have two distinct real solutions. If it's zero, you have exactly one real solution (a repeated root), and if it's negative, the solutions are complex or involve imaginary numbers.

Imaginary Number

An imaginary number arises when the square root of a negative number is encountered. The imaginary unit is represented by $i$, which is defined by the property:

$ i^2 = -1 $

This concept becomes essential when dealing with equations where the discriminant $b^2 - 4ac$ is negative. In such cases, the solutions will not be real numbers, as real numbers do not satisfy the condition of a negative square root. Instead, they include an imaginary component. Imaginary numbers, combined with real numbers, form complex numbers, expanding our numerical system to solve more comprehensive mathematical problems.

Solutions in a+bi Form

When calculating roots for a quadratic equation with a negative discriminant, you end up with complex solutions. The solutions will be in the form $a + bi$, where:

$a$ is the real part and represents the real number component.
$b$ is the imaginary part and $i$ symbolizes the imaginary unit.

These complex solutions are crucial for expressing numbers that can't be resolved within the realm of real numbers. For instance, after simplifying an expression like $\sqrt{-8}$ from a quadratic solution, you would denote this as $\sqrt{8}i$. The final result might look like $\frac{2}{3} \pm \frac{2\sqrt{2}}{3}i$, representing two unique solutions due to the $\pm$. This form ensures every quadratic equation, regardless of the discriminant, can have a valid solution.

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91影视

Solve each equation. Write the answer in bi or a \(+\) bi form. $$3 x^{2}-4 x+2=0$$

Short Answer

Step by step solution

Identify a, b, and c

Apply the quadratic formula

Simplify the expression under the square root

Handle the negative under the square root

Simplify final results

Key Concepts

Quadratic Equation

Quadratic Formula

Imaginary Number

Solutions in a+bi Form

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