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Chapter 1: Problem 67

Solve each equation in the complex number system. $$ 25 x^{2}-10 x+2=0 $$

Short Answer

Expert verified
The solutions are \(x = \frac{1}{5} + \frac{1}{5}i\) and \(x = \frac{1}{5} - \frac{1}{5}i\).

Step by step solution

01

Identify coefficients

For the quadratic equation in the form \[ax^2 + bx + c = 0\]Identify the coefficients: \[a = 25, \, b = -10, \, c = 2\]
02

Use the quadratic formula

The quadratic formula is given by: \[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]Substitute the coefficients into the formula.
03

Calculate the discriminant

Calculate the discriminant \[D = b^2 - 4ac\]Substitute \(b = -10\), \(a = 25\), and \(c = 2\): \[D = (-10)^2 - 4(25)(2) = 100 - 200 = -100\]The discriminant is \(-100\), which is negative.
04

Simplify under the square root

Since the discriminant is negative, the solutions will be complex. Write the square root of the discriminant as: \[\sqrt{-100} = \sqrt{100 \cdot -1} = 10i\]Here, \(i\) is the imaginary unit where \(i^2 = -1\).
05

Solve using the quadratic formula

Substitute \(D = -100\) back into the quadratic formula: \[x = \frac{-(-10) \pm 10i}{2(25)} = \frac{10 \pm 10i}{50} = \frac{10}{50} \pm \frac{10i}{50}\]Simplify the fractions: \[x = \frac{1}{5} \pm \frac{1}{5}i\]

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Key Concepts

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

Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations of the form \(ax^2 + bx + c = 0\). It is expressed as:
\[x = \frac{-b \pm \sqrt{b^2-4ac}}{2a}\]
Here:
  • \textbf{a}: Coefficient of the quadratic term \(x^2\)
  • \textbf{b}: Coefficient of the linear term \(x\)
  • \textbf{c}: Constant term
The formula provides solutions as long as you know the coefficients \(a\), \(b\), and \(c\).
Simply substitute these values into the formula to find the solutions for \(x\). This formula covers all types of quadratic equations, whether the solutions are real or complex. It works even when the discriminant \(b^2 - 4ac\) is negative.
Complex Numbers
When solving quadratic equations, sometimes you'll encounter a negative discriminant. This means the equation has no real solutions. Instead, it has complex solutions.
Complex numbers have the form \(a + bi\), where:
  • \textbf{a}: Real part
  • \textbf{b}: Imaginary part
  • \textbf{i}: Imaginary unit where \(i^2 = -1\)
For example, \(\sqrt{-100} = 10i\). This is because \(-100\) can be written as \(100 \times -1\). Hence, \(\sqrt{-100} = \sqrt{100} \times \sqrt{-1} = 10i\). Complex solutions come in conjugate pairs. If one solution is \(\frac{1}{5} + \frac{1}{5}i \), the other is \(\frac{1}{5} - \frac{1}{5}i\).
Discriminant
The discriminant \(D = b^2 - 4ac\) decides the nature of the roots of a quadratic equation:
  • \textbf{If \(D > 0\)}: The equation has two distinct real roots.
  • \textbf{If \(D = 0\)}: The equation has exactly one real root.
  • \textbf{If \(D < 0\)}: The equation has two complex roots.
In our example, the discriminant is \(D = -100\). Because \(D < 0\), it means the equation has complex roots. Introducing the concept of the discriminant helps predict the types of solutions we can expect from a quadratic equation, even before fully solving it.

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