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Chapter 1: Problem 31

Solve each equation by the Square Root Method. $$ x^{2}=25 $$

Short Answer

Expert verified
The solutions are \( x = 5 \) and \( x = -5 \).

Step by step solution

01

Understand the Square Root Method

The Square Root Method is used to solve quadratic equations of the form \( x^{2} = k \) by taking the square root of both sides of the equation.
02

Isolate the Square Term

In the given equation \( x^{2} = 25 \), the square term \( x^{2} \) is already isolated on one side. This means we can directly proceed to the next step.
03

Take the Square Root of Both Sides

Take the square root of both sides of the equation: \[ \sqrt{x^{2}} = \sqrt{25} \] This simplifies to: \[ x = \pm 5 \]
04

Determine the Solutions

The solutions to the equation are \( x = 5 \) and \( x = -5 \). This is because both \( (5)^{2} = 25 \) and \( (-5)^{2} = 25 \).

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Key Concepts

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

Quadratic Equations
To start with, let's understand what quadratic equations are. Quadratic equations are mathematical expressions that follow the general form: \[ ax^2 + bx + c = 0 \]Here, 'a', 'b', and 'c' are constants, and 'x' represents the variable. The key characteristic of a quadratic equation is the term 'x^2', which is a square term. These equations usually have two solutions because of their parabolic nature. The solutions can be real or complex numbers.
One of the ways to solve quadratic equations is by using the Square Root Method. This is applicable when the equation is in the form of \[ x^2 = k \].
Isolating Square Terms
Before we solve a quadratic equation using the Square Root Method, we need to isolate the square term.
  • For an equation in the form \[ax^2 + bx + c = 0\], first move the constant term and other terms involving 'x' to the opposite side.
  • Then, make sure the square term is alone on one side of the equation.
    This often involves dividing the entire equation by the coefficient of the square term, so that it looks like \[x^2 = k \].
    For example, if we start with
    \[2x^2 - 8 = 0\], we would add 8 to both sides to get \[2x^2 = 8\], and then divide by 2 to simplify it: \[x^2 = 4\].
    Once the equation is in this form, we can proceed to take the square root.
Taking Square Roots
Now that we have isolated the square term to have an equation in the form of \[x^2 = k\], we can take the square root of both sides.
Remember, the square root has both a positive and a negative solution.
  • For the equation \[x^2 = 25\]:
  • Taking the square root of both sides gives us
    \[\begin{align*}\text{Left Side:}& \ \ \ \text{Right Side:}& \begin{tabular}{rcl}\begin{array}{l}\text{Simplify to:}\end{array} & \begin{array}{l}\x \pm 5\end{array}\end{tabular}\}& \text{root}\]
    This means that both \[5^2 = 25\] and
    \[(-5)^2 = 25\].
Solving Equations
Once you have taken the square root of both sides, you'll get the potential solutions.
For equations of the form \[x^2 = k\]:
  • We typically write the solutions as
    \[x = \pm \sqrt{k} \]
  • This captures both the positive and negative roots.
    In the specific example given, \[x^2 = 25\]:
    \[\therefore x = 5 \quad \text{or} \quad x = -5\]
    • Sometimes, for equations with more terms, ensure you validate each solution by plugging back into the original equation.
      • This prevents errors especially in more complex situations.
    Understanding the methods increases confidence in solving such equations.

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