Match the equation in Column I with its solution(s) in Column II. x^2 = 25

Verified step by step guidance

1. Recognize that the equation \(x^2 = 25\) is a quadratic equation in the form \(x^2 = c\), where \(c\) is a constant.
2. To solve \(x^2 = 25\), take the square root of both sides of the equation. Remember that taking the square root of a number results in both a positive and a negative solution.
3. This gives us \(x = \pm \sqrt{25}\).
4. Calculate the square root of 25, which is 5.
5. Thus, the solutions to the equation are \(x = 5\) and \(x = -5\).

Key Concepts

Quadratic Equations

A quadratic equation is a polynomial equation of the form ax^2 + bx + c = 0, where a, b, and c are constants and a ≠ 0. The solutions to these equations can be found using various methods, including factoring, completing the square, or applying the quadratic formula. In the case of x^2 = 25, it can be rearranged to the standard form and solved accordingly.

Square Roots

Square roots are the values that, when multiplied by themselves, yield the original number. For example, the square root of 25 is 5, since 5 * 5 = 25. When solving equations like x^2 = 25, taking the square root of both sides leads to two potential solutions: x = 5 and x = -5, reflecting the property that both positive and negative values can satisfy the equation.

Solution Sets

A solution set is the collection of all possible solutions to an equation. For quadratic equations, the solution set can include one solution (a repeated root), two distinct solutions, or no real solutions at all. In the case of x^2 = 25, the solution set is {5, -5}, indicating that both values satisfy the original equation.