Question

If \(x^2=25\), which of the following is true? $$ \begin{aligned} & \bigcirc x=5 \\ & \bigotimes x=-5 \\ & \hdashline x=\pm 5 \\ & \text { idon't know. } \end{aligned} $$

Short answer

Therefore, the correct answer is \(x=\pm 5\).

Step-by-step solution

Identify the equation

The equation to solve here is \(x^2=25\), which is a standard form of a quadratic equation.

Square root both sides

To solve for \(x\), both sides of the equation could be square rooted. The equation now becomes \(x=\sqrt{25}\) which has two solutions: +5 or -5.

Choose Answer

From the options given, the symbol \(\hdashline\) indicates that both +5 and -5 are correct solutions. Therefore, \(x = \pm 5\) is the right choice.

Key concepts

Solving Quadratic Equations

Quadratic equations are a special type of polynomial equation where the highest exponent of the variable, usually represented as \( x \), is 2. In other words, quadratic equations take the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants. The goal is to find the value or values of \( x \) that make the equation true. The equation \( x^2 = 25 \) is a perfect example of a quadratic equation where the coefficients \( b \) and \( c \) are zero, simplifying it even more.

• The first step in approaching such equations is identifying their form and simplifying them, if possible.
• Next, apply algebraic methods that fit the configuration of the equation, such as factoring, using the quadratic formula, or, like in our case, using square roots.
• Finding solutions is essential since quadratic equations often have two solutions due to the nature of the squared term.

By understanding the basic structure of quadratic equations, you'll be better equipped to tackle them efficiently and accurately.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. To solve \( x^2 = 25 \), we move to take the square root of both sides. This is a key technique for handling simple quadratic equations and is rooted in the inverse operations concept.

• Taking the square root of a squared variable, \( x^2 \), gives us \( x \). However, it's crucial to remember that the square root process involves considering both the positive and negative roots.
• Thus, from \( x^2 = 25 \), we derive \( x = \pm 5 \).

This indicates there are two solutions for every perfect square, leading to a deeper understanding of quadratic equations. Keep in mind that once you recognize this pattern, many quadratic solutions become intuitive.

Algebraic Solutions

An algebraic solution can be described as solving equations using algebraic manipulations rather than numerical or graphical methods. For polynomial equations like quadratics, this often involves rearranging terms, taking roots, or using specialized formulas.

• Using algebraic techniques, one can often reduce a problem to a simpler form that's easier to solve.
• In the case of \( x^2 = 25 \), applying the square root directly leads us to the solutions without additional steps.
• Understanding symmetry in algebra helps visualize why \( x = \pm 5 \) is true: zeroing the squared term requires equal and opposite values on the number line.

Such solutions offer a consistent and reliable method for solving quadratic equations across many contexts, whether they appear in pure math exercises or real-world applications.