Problem 334 - Elementary Algebra

Question

Solve. \(625=x^{2}\)

Step-by-Step Solution

Answer

The solutions are \(x = 25\) and \(x = -25\).

Step 1: Understand the equation

The given equation is a simple quadratic equation: \[625 = x^2\] We need to find the value of \(x\) that satisfies this equation.

Step 2: Take the square root of both sides

To isolate \(x\), take the square root of both sides of the equation: \[\sqrt{625} = \sqrt{x^2}\] Remember, taking the square root introduces both a positive and a negative solution.

Step 3: Simplify the square roots

Simplify both square roots: \[\sqrt{625} = 25\] and \[\sqrt{x^2} = |x|\] This gives us: \[25 = |x|\]

Step 4: Solve for positive and negative x

Since the absolute value of \(x\) is 25, we have two possible solutions: \[ x = 25 \] and \[ x = -25 \]

Key Concepts

quadratic equation

A quadratic equation is a type of polynomial equation of the form \[ ax^2 + bx + c = 0 \]where \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). In the given exercise, the quadratic equation is written in a different format: \[ 625 = x^2 \]Quadratic equations can be solved using various methods like factoring, using the quadratic formula, or by taking the square root (if the equation is in the form \(x^2 = k\)). In our exercise, we solve it by taking the square root.

square root

The square root of a number \(k\) is a value \(x\) such that \( x \times x = k \)The symbol for square root is \( \sqrt{} \). In the exercise, we take the square root of both sides of the equation: \[ \sqrt{625} = \sqrt{x^2} \]This helps to isolate \(x\) by removing the exponent. Don't forget that taking the square root gives both positive and negative values since \( (25 \times 25 = 625) \)and \( (-25 \times -25 = 625)\).So, \[ \sqrt{625} = ±25 \]and \[ \sqrt{x^2} = |x| \]With steps like these, remember to consider both positive and negative solutions.

absolute value

The absolute value of a number is its distance from zero on the number line, regardless of direction. It is always non-negative. The symbol for absolute value is two vertical bars: \(|x|\)For example, \(|25| = 25\)and \(|-25| = 25\)In the exercise, \[ 25 = |x| \]means that \(x\) is either \(25\)or \(-25\)The absolute value ensures that both these values are considered solutions.

positive and negative solutions

When solving \( x^2 = k \)by taking the square root, always remember that there are both a positive and a negative solution. This is due to the property of squares, where both positive and negative values result in positive squares:\[ (25 \times 25 = 625) \]and \[ (-25 \times -25 = 625) \]In the given equation \( 625 = x^2 \)taking the square root of both sides yields \[ x = \pm25 \]Hence, the two solutions to the equation are \( x = 25 \)and \( x = -25 \). Always consider both possibilities to avoid missing any solutions.