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Chapter 1: Problem 36

Find all real solutions of the equation. \(x^{2}-6 x+1=0\)

Short Answer

Expert verified
The real solutions are \( x = 3 + 2\sqrt{2} \) and \( x = 3 - 2\sqrt{2} \).

Step by step solution

01

Identify the equation type

The given equation is a quadratic equation of the form \( ax^2 + bx + c = 0 \), where \( a = 1 \), \( b = -6 \), and \( c = 1 \).
02

Write the quadratic formula

To find the real solutions, we use the quadratic formula: \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \).
03

Calculate the discriminant

First, calculate the discriminant \( b^2 - 4ac \). Substituting in the values, \( (-6)^2 - 4 \times 1 \times 1 = 36 - 4 = 32 \).
04

Compute the square root of the discriminant

Calculate \( \sqrt{32} \), which is \( 4\sqrt{2} \). The fact that the discriminant is positive indicates two real solutions.
05

Substitute into the quadratic formula

Substitute \( b = -6 \), \( \sqrt{32} = 4\sqrt{2} \), and \( a = 1 \) into the quadratic formula: \[ x = \frac{-(-6) \pm 4\sqrt{2}}{2 \times 1} = \frac{6 \pm 4\sqrt{2}}{2} \].
06

Simplify the expression

Simplify the solutions: \[ x = \frac{6 + 4\sqrt{2}}{2} = 3 + 2\sqrt{2} \]\[ x = \frac{6 - 4\sqrt{2}}{2} = 3 - 2\sqrt{2} \].

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Key Concepts

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

Understanding the Discriminant
In the world of quadratic equations, the discriminant is a key component. It is a part of the quadratic formula, and it helps us determine the nature of the solutions. The discriminant is given by the formula \( b^2 - 4ac \), where \( b \) and \( c \) are coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).
The value of the discriminant tells us:
  • If it is positive, there are two distinct real solutions.
  • If it is zero, there is exactly one real solution (also known as a repeated root).
  • If it is negative, there are no real solutions; instead, there are two complex solutions.
In our original exercise, the discriminant was calculated as \( 32 \). Since this is a positive number, we knew that there are two real solutions for the equation.
The Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It can be used to find the roots (solutions) of any quadratic equation and is written as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
This essential formula provides a method to find the solutions when factoring is not possible. It includes several components:
  • \(-b\) symbolizes the opposite of the coefficient \(b\).
  • The square root of the discriminant \( \sqrt{b^2 - 4ac} \) dictates the nature of the solutions.
  • The denominator \(2a\) normalizes the values of the numerator.
When using the quadratic formula, substitute the coefficients \( a \), \( b \), and \( c \) from the original equation. For our equation \( x^2 - 6x + 1 = 0 \), we found the solutions by substituting values into the formula, leading to the expressions \( 3 + 2\sqrt{2} \) and \( 3 - 2\sqrt{2} \).
Finding Real Solutions
Real solutions of a quadratic equation are simply the \(x\)-values that satisfy the equation and can be viewed on a graph intersecting the \(x\)-axis. In the case of real solutions, they are the points where the parabola, represented by the quadratic equation, touches or cuts through the \(x\)-axis.
When we determine the real solutions:
  • We are solving for \(x\) in \(ax^2 + bx + c = 0\).
  • These solutions can either be different or the same, depending on the value of the discriminant.
In our example, since the discriminant was positive, we found two real solutions: \(3 + 2\sqrt{2}\) and \(3 - 2\sqrt{2}\). These solutions tell us the points of intersection of the quadratic equation \(x^2 - 6x + 1 = 0\) with the \(x\)-axis on a graph.

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