Problem 48 - Elementary Algebra

Question

Solve using the Square Root Property. \((y-4)^{2}=64\)

Step-by-Step Solution

Answer

The solutions are \(y = 12\) and \(y = -4\).

Step 1 - Isolate the squared term

The equation is already set with the squared term isolated: \( (y-4)^{2} = 64 \).

Step 2 - Apply the Square Root Property

To solve for \(y\), take the square root of both sides of the equation: \[ \sqrt{(y-4)^{2}} = \pm \sqrt{64} \].

Step 3 - Simplify the square roots

Simplify both sides to get: \[ y - 4 = \pm 8 \]. This means \(y - 4 = 8\) or \(y - 4 = -8\).

Step 4 - Solve for y in both cases

For \(y - 4 = 8\), add 4 to both sides to get \(y = 12\). \ For \(y - 4 = -8\), add 4 to both sides to get \(y = -4\).

Step 5 - Write the final solution

The values for \(y\) that satisfy the equation are \(y = 12\) and \(y = -4\).

Key Concepts

solving equations

To solve equations, especially those involving algebraic expressions, it's important to follow systematic steps. Start by isolating the variable you're solving for. In the given exercise, we first noticed that the equation was already set up with the squared term isolated \[ (y - 4)^{2} = 64 \]. This made the next steps straightforward.
When isolating, you might need to perform operations like addition, subtraction, multiplication, or division on both sides of the equation. Always aim to simplify the equation to make solving easier.

factoring

Factoring is a key technique used in solving many algebraic equations. It involves breaking down an expression into simpler components, or 'factors'. In the context of the exercise, factoring wasn't directly needed because the expression \[ (y - 4)^{2} = 64 \] was already in a simplified form.
However, understanding factoring is essential for solving quadratic equations. For example, if you have \[ x^{2} + 5x + 6 = 0 \], you can factor it into \[ (x + 2)(x + 3) = 0 \]. This then allows you to solve for \[ x \] by setting each factor to zero.

algebraic expressions

Algebraic expressions are combinations of numbers, variables, and arithmetic operations. In the problem provided, \[ (y - 4)^{2} = 64 \] is an algebraic expression.
Working with these expressions often involves simplifying them to make equations easier to solve. Here, we applied the Square Root Property to both sides of the equation, simplifying it to \[ y - 4 = \pm 8 \]. Knowing how to manipulate and simplify such expressions is critical in all of algebra.

quadratic equations

Quadratic equations are equations of the form \[ ax^{2} + bx + c = 0 \]. The given exercise \[ (y - 4)^{2} = 64 \] can be viewed as a quadratic equation in disguise.
Solving quadratic equations often involves factoring, using the quadratic formula, or completing the square. Here, we utilized the Square Root Property, simplified to \[ y - 4 = \pm 8 \], and then solved each case separately, leading to the solutions \[ y = 12 \] and \[ y = -4 \].
Mastering these methods is crucial for success in algebra.