Problem 22 - Algebra 1: Concepts and Skills

Question

Solve the equation or write no real solution. Write the solutions as integers if possible. Otherwise, write them as radical expressions. $$ y^{2}=15 $$

Step-by-Step Solution

Answer

The solutions to the equation \( y^{2}=15 \) are \( y = +\sqrt{15} \) and \( y = -\sqrt{15} \).

Step 1 - Identify the parameters

The equation is \( y^{2}=15 \). There is only one variable y. The number 15 is a constant.

Step 2 - Apply square root on both sides

To solve for y, the square root of both sides has to be taken. This yields: \( y = +\sqrt{15} \) and \( y = -\sqrt{15} \). Remember, a positive and negative solution exists because squaring any real number (both positive and negative) will result in a positive result.

Step 3 - Simplify answer (if possible)

The square root of 15 cannot be further simplified into an integer, so the final answers are \( y = +\sqrt{15} \) and \( y = -\sqrt{15} \) respectively. These are in radical expressions as asked for in the problem statement.

Key Concepts

Square Roots

The square root is a mathematical concept that involves finding a number that, when multiplied by itself, gives the original number. In the context of quadratic equations like \( y^2 = 15 \), taking the square root allows us to solve for \( y \). This is because the square root of \( y^2 \) is \( y \) itself. To solve the equation, both the positive and negative square roots of the constant on the other side of the equation need to be considered, giving solutions:

• \( y = +\sqrt{15} \)
• \( y = -\sqrt{15} \)

It's important to remember that the square root of any positive number is actually two numbers, one positive and one negative. This is due to the fact that both \( a^2 \) and \( (-a)^2 \) result in the same positive number. Hence, always take both solutions into account when solving quadratic equations.

Radical Expressions

Radical expressions are mathematical expressions that contain a square root, cube root, or any higher-order root. In our specific example, \( \sqrt{15} \) is a radical expression.

• The radical sign, \( \sqrt{} \), indicates that you are finding a root of a number.
• The number under the radical sign is called the radicand.
• Our radicand here is 15, which is not a perfect square.

When the radical's value cannot be simplified to an integer, the expression remains in its radical form, such as \( \sqrt{15} \). Radical expressions are exact and often preferred when dealing with roots of numbers that are not perfect squares, as they provide more precise values than decimals.

Real Solutions

Real solutions are solutions to an equation that can be found on the real number line. For quadratic equations like \( y^2 = 15 \), real solutions are the values of \( y \) that satisfy the equation. The unique feature of quadratic equations is that they may have two real solutions, one positive and one negative.

• The equation \( y^2 = 15 \) yields real solutions \( y = +\sqrt{15} \) and \( y = -\sqrt{15} \).
• Both solutions are real because \( \sqrt{15} \) is a real number.
• Quadratic equations can sometimes have "no real solutions," which happens if an equation's solution involves the square root of a negative number, leading to complex or imaginary numbers.

In this case, since we have a positive constant 15, \( \sqrt{15} \) provides real solutions without involving imaginary numbers.