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Chapter 0: Problem 42

Solve by completing the square. $$2 x^{2}-4 x+3=0$$

Short Answer

Expert verified
The equation has no real solutions because the square is negative.

Step by step solution

01

Move the constant term to the other side

Begin by moving the constant term (3) to the right side of the equation. This leads to the equation: \[ 2x^2 - 4x = -3 \]
02

Factor out the coefficient of the quadratic term

The equation has a quadratic term with a coefficient of 2. Factor out 2 from the left side: \[ 2(x^2 - 2x) = -3 \]
03

Complete the square inside the parentheses

To complete the square, take half of the coefficient of x, square it, and add it inside the parentheses. Here, half of -2 is -1, and its square is 1.Add and subtract 1 inside the parentheses:\[ 2(x^2 - 2x + 1 - 1) = -3 \]\[ 2((x - 1)^2 - 1) = -3 \]
04

Simplify and adjust the equation

Distribute the 2 back into the expression:\[ 2((x - 1)^2) - 2 imes 1 = -3 \]\[ 2(x - 1)^2 - 2 = -3 \]
05

Isolate the perfect square term

Add 2 to both sides of the equation to isolate the perfect square:\[ 2(x - 1)^2 = -3 + 2 \]\[ 2(x - 1)^2 = -1 \]
06

Solve for the variable

Divide the whole equation by 2 to solve for \((x - 1)^2\):\[ (x - 1)^2 = -\frac{1}{2} \]Since \((x - 1)^2\) is equal to a negative number, there is no real solution, and this suggests complex roots.

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Key Concepts

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

Quadratic Equations
Quadratic equations take the form \(ax^2 + bx + c = 0\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable. These equations describe a parabola when graphed, and they can open upward or downward depending on the sign of \(a\). Solving these equations is an essential skill in mathematics, especially in algebra.
  • The simplest method to solve quadratic equations is by factoring when the quadratic can be easily decomposed into binomials.
  • If factoring is not straightforward, methods such as completing the square or using the quadratic formula become useful.
When performing operations like completing the square, the goal is to express the quadratic in a form that makes it easier to solve for \(x\), such as \((x - p)^2 = q\). Understanding quadratic equations is fundamental to exploring more advanced mathematical concepts such as functions and calculus.
Complex Roots
Complex roots arise when a quadratic equation cannot be solved using real numbers. This happens when the value under the square root (discriminant) in the quadratic formula becomes negative. The quadratic formula is given by:\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]If \(b^2 - 4ac < 0\), the roots will be complex, taking the form \(p \pm qi\), where \(i\) represents the imaginary unit, \(\sqrt{-1}\).
  • Complex numbers are represented as \(a + bi\), where \(a\) and \(b\) are real numbers.
  • Imaginary numbers allow the extension of real numbers and solve equations that otherwise have no solution in the real number system.
In our problem, after completing the square, we encounter a negative in the square, indicating the presence of complex roots. This means the original quadratic equation has no real solutions, but rather solutions that include imaginary numbers.
Solving Equations
Solving equations involves finding the values for the variables that make the equation true. For quadratic equations, several strategies can be applied depending on the characteristics of the equation.
  • Factoring works best when the equation is easily decomposable into simpler expressions.
  • Completing the square is a versatile method, particularly useful when the quadratic does not factor neatly.
  • The quadratic formula provides a direct way to solve any quadratic equation, which is especially useful when other methods become cumbersome.
In the problem solver's exercise, completing the square was used effectively to isolate a perfect square, even though it led to complex roots. Solving through this method often involves manipulating the equation step-by-step to transform it into a more solvable form. This includes moving terms, factoring, and eventually solving for the roots, whether real or complex.

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Most popular questions from this chapter

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