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Chapter 1: Problem 63

Solve the equation for \(x\). $$a^{2} x^{2}+2 a x+1=0 \quad(a \neq 0)$$

Short Answer

Expert verified
The solution is \(x = -\frac{1}{a}\).

Step by step solution

01

Identify the form of the equation

The given equation is of the form \(a^2x^2 + 2ax + 1 = 0\), which is a quadratic equation.
02

Recognize the coefficients

In the quadratic equation \(ax^2 + bx + c = 0\), identify the coefficients: \(a = a^2\), \(b = 2a\), and \(c = 1\).
03

Use the quadratic formula

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) is employed here. Plug in the values of \(a\), \(b\), and \(c\).
04

Calculate the discriminant

Calculate the discriminant \(b^2 - 4ac\) which is \((2a)^2 - 4(a^2)(1) = 4a^2 - 4a^2 = 0\).
05

Solve using the discriminant

Since the discriminant is 0, there is one solution: \(x = \frac{-b}{2a} = \frac{-2a}{2a^2} = -\frac{1}{a}\).

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Key Concepts

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

Understanding the Discriminant
In a quadratic equation of the form \( ax^2 + bx + c = 0 \), the discriminant is given by the expression \( b^2 - 4ac \). The discriminant plays a key role in determining the nature of the roots of the quadratic equation.

Here are the key points to remember about the discriminant:
  • If the discriminant is positive, the quadratic equation has two distinct real roots.
  • If the discriminant is zero, the equation has exactly one real root, also known as a repeated or double root.
  • If the discriminant is negative, there are no real roots. Instead, the roots are complex and occur as conjugate pairs.
In this exercise, the discriminant is calculated as follows: \( b^2 - 4ac = (2a)^2 - 4(a^2)(1) = 4a^2 - 4a^2 = 0 \).

The result being zero indicates that the quadratic equation \( a^2 x^2 + 2ax + 1 = 0 \) has exactly one real solution.
Using the Quadratic Formula
The quadratic formula is an essential tool for solving any quadratic equation of the form \( ax^2 + bx + c = 0 \). The formula is given by:
  • \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
This formula allows us to calculate the roots directly by plugging in the values of the coefficients \( a \), \( b \), and \( c \). Here鈥檚 how it works step-by-step:
  • First, calculate the discriminant, \( b^2 - 4ac \).
  • Then, take the square root of the discriminant.
  • Finally, use the formula to find the values of \( x \) by substituting the known values of \( b \), \( \pm \sqrt{b^2 - 4ac} \), and \( a \).
In this problem, once the discriminant value of zero was found, it simplified our work, allowing us to find the solution as: \( x = \frac{-2a}{2a^2} = -\frac{1}{a} \), without needing the "plus or minus" operation.
Step-by-Step Mathematical Solution
To solve the quadratic equation \( a^2 x^2 + 2ax + 1 = 0 \) using mathematical solution steps, it's important to methodically apply the quadratic formula and related concepts.

Here are the detailed steps:
  • First, identify that the equation is quadratic. It is in the form \( a^2 x^2 + 2ax + 1 = 0 \).
  • Next, recognize the coefficients necessary for using the quadratic formula. We notice: \( a = a^2 \), \( b = 2a \), and \( c = 1 \).
  • With the coefficients identified, use the quadratic formula. Plug \( a \), \( b \), and \( c \) into \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
  • Calculate the discriminant \( b^2 - 4ac = 0 \), indicating one real root.
  • Finalize the solution by simplifying: \( x = \frac{-2a}{2a^2} = -\frac{1}{a} \).
Through these steps, each part of the formula and every calculation decision contribute to determining \( x \). The outcome illustrates how the mathematical steps intertwine to yield a logical solution to the problem.

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