Question

Solve \(x^2=25\). A. \(x=5\) B. \(x=-5\) C. \(x= \pm 5\) D. \(x=12.5\)

Short answer

C. \(x= \pm 5\)

Step-by-step solution

Identify the given equation

The given equation is \(x^2 = 25\)

Understand the square root property

To solve \(x^2 = 25\), take the square root of both sides of the equation. Note that taking the square root of both sides introduces both positive and negative roots.

Solve for x

Taking the square root of both sides: \(\sqrt{x^2} = \sqrt{25}\). This simplifies to \(x = \pm 5\).

Select the correct answer

Among the given choices, the correct answer is C. \(x= \pm 5\).

Key concepts

Square Root Property

When you need to solve an equation like \(\textstyle{x^2 = 25}\), you can use the square root property. This property states that for any number \(\textstyle{a}\), if \(\textstyle{x^2 = a}\), then \(\textstyle{x = \pm \sqrt{a}}\). In simpler terms, taking the square root of both sides of an equation will help you find the value of \(\textstyle{x}\).

Remember, the square root is the value that, when multiplied by itself, gives the original number. For example, \(\textstyle{\text{5} \times \text{5} = 25}\), so the square root of \(\textstyle{25}\) is \(\textstyle{5}\). But there's another important part to remember!

Positive and Negative Roots

When you take the square root of a number, you get two possible answers: one positive and one negative. This is because both \(\textstyle{5 \times 5 = 25}\) and \(\textstyle{-5 \times -5 = 25}\). Therefore, for \(\textstyle{x^2 = 25}\), \(\textstyle{x}\) can be both \(\textstyle{5}\) and \(\textstyle{-5}\).

In mathematical notation, this is written as \(\textstyle{x = \pm 5}\). The \(\textstyle{\pm}\) sign means 'plus or minus,' indicating both positive and negative roots.

Simplifying Radicals

Radicals are symbols used to represent roots of numbers. In our equation \(\textstyle{x^2 = 25}\), the radical involved is the square root. Simplifying a radical means finding its simplest form. For example, \(\textstyle{\text{√25} = 5}\) because \(\textstyle{5}\) times \(\textstyle{5}\) equals \(\textstyle{25}\).

When simplifying radicals, always look for perfect squares (like 4, 9, 16) under the radical sign. These make it easier to simplify to whole numbers.

Combining all these, you understand that solving \(\textstyle{x^2 = 25}\) involves recognizing the square root property, remembering the positive and negative roots, and simplifying the radical to get the final answer: \(\textstyle{x = \pm 5}\).