Question

Solve the equation or write no solution. Write the solutions as integers if possible. Otherwise write them as radical expressions. $$b^{2}=64$$

Short answer

The solution for the equation \( b^{2}=64 \) is \( b = -8, 8 \)

Step-by-step solution

Identify the type of equation

The given equation \( b^2 = 64 \) is a simple square equation.

Take the square root of both sides

In order to isolate b, we need to take the square root of both sides of the equation. Doing this, we get \( b = \sqrt{64} \)

Evaluate

We must remember that we have both positive and negative roots for the equation. So the value for b will be \( b = -8, 8 \)

Key concepts

Square Root Method

The square root method is a straightforward technique for solving quadratic equations of the form (x^2 = c), where (c) is a constant. The process involves taking the square root of both sides of the equation to solve for (x). This method can quickly find the roots of a quadratic equation, provided that one side of the equation is a perfect square, as seen with (b^2 = 64).

However, it's imperative to remember that squaring is a reversible operation, and thus there are always two possible solutions: one positive and one negative. In the case of (b^2 = 64), you would calculate (b = \[ +\sqrt{64} \]) and (b = \[ -\sqrt{64} \]) to get the two integer solutions: (+8) and (-8).

Radical Expressions

Radical expressions involve roots, such as square roots, cube roots, and so on. These expressions play a significant role when solving equations where the variable is under a radical. Students must remember that roots come in pairs — specifically, taking the square root of a number gives both a positive and negative value, because squaring either of these would return the original number.

For instance, in the equation (b^2 = 64), the square root method requires us to express (b) as a radical expression if necessary. Because 64 is a perfect square, we can express (b) as the integers (8) and (-8) following the radical symbol. However, not all solutions may be neat integers; sometimes, students will encounter non-perfect squares that result in a radical expression remaining in the solution.

Quadratic Formula

The quadratic formula is a one-stop solution for any quadratic equation of the form (ax^2 + bx + c = 0). If factoring or the square root method are not viable, or if the equation does not simplify to a basic form, the quadratic formula can be applied. It states that the solutions to the quadratic equation are ((x = \[(-b \pm \sqrt{b^2 - 4ac})/(2a)\])).

Despite not being necessary for simple equations like (b^2 = 64), understanding the quadratic formula is vital since it provides solutions even when other methods fail or are cumbersome. Additionally, the discriminant (\b^2 - 4ac) portion of the formula determines the nature of the roots, whether they are real numbers or complex, and can lead to radical expressions when not a perfect square.

Integer Solutions

When solving equations, one might find different types of solutions: integers, fractions, irrationals, or even complex numbers. Integer solutions are whole numbers, both positive and negative, including zero. Identifying integer solutions can simplify the expression of the equation's solution and are particularly satisfying when you can provide a clear, simple answer.

In the context of a quadratic equation like (b^2 = 64), integrity of the solution is confirmed when the square root of the constant is a perfect square, leading to the integer solutions (8) and (-8). However, if the constant were not a perfect square, the solutions might not be integers but could be irrational and expressed as radical expressions.