Solve each equation in Exercises 83–108 by the method of your choice. 3x^2 = 60

Verified step by step guidance

1. Start by isolating the quadratic term. Divide both sides of the equation by 3 to simplify: \( x^2 = \frac{60}{3} \).
2. Simplify the right side of the equation: \( x^2 = 20 \).
3. To solve for \( x \), take the square root of both sides of the equation: \( x = \pm \sqrt{20} \).
4. Simplify the square root if possible. Note that \( \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \times \sqrt{5} = 2\sqrt{5} \).
5. Thus, the solutions are \( x = 2\sqrt{5} \) and \( x = -2\sqrt{5} \).

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. In this case, the equation 3x^2 = 60 can be rearranged into standard form by moving all terms to one side, resulting in 3x^2 - 60 = 0. Understanding how to manipulate and solve quadratic equations is essential for finding the values of x.

Factoring

Factoring is the process of breaking down an expression into simpler components, or factors, that when multiplied together yield the original expression. For the equation 3x^2 - 60 = 0, one can factor out the common term, leading to 3(x^2 - 20) = 0. This technique simplifies the equation and makes it easier to solve for x.

Square Roots

The square root of a number is a value that, when multiplied by itself, gives the original number. In solving the equation 3(x^2 - 20) = 0, we can isolate x^2 = 20 and then take the square root of both sides. This step is crucial as it leads to the final solutions for x, which can be both positive and negative.