Problem 3 - Algebra 1

Question

Is 36 a solution of \(\sqrt{x}=-6 ?\) Why or why not?

Step-by-Step Solution

Answer

No, 36 is not the solution to the equation \(\sqrt{x}=-6\), because a square root cannot yield a negative number.

Step 1: Understand Properties of Square Root

We should recall that square root of any real number is always positive or zero but never negative. This is because when you square any real number, negative or positive, you always get a positive result or zero, never a negative result. So, the equation \(\sqrt{x}=-6\) does not have any real number solutions.

Step 2: Substitute the Given Value

In the next step, substitute x = 36 into the equation \(\sqrt{x} = -6\) to see if both sides are equal, despite knowing already that it can't be a solution. If the results are equal, then 36 is a solution. Otherwise, it's not a solution.

Step 3: Check if Both Sides are Equal

Calculating the square root of 36, we get \( \sqrt{36} = 6 \). Now compare both sides now and see that 6 is not equal to -6. Therefore, the value 36 is not a solution of the equation, \(\sqrt{x}=-6\).

Key Concepts

Non-negative Numbers

Non-negative numbers include all the numbers that are greater than or equal to zero. These are the numbers that do not have a negative sign. They form an important subset of real numbers and serve as the foundation for understanding many mathematical operations, including taking square roots.

• For any real number squared, the result is always a non-negative number.
• This means numbers like 0, 1, 2, and even fractions like 1/2 are non-negative.

When dealing with square roots, it's especially important to note that non-negative numbers are the only numbers that can have real square roots. This means \(\sqrt{x}\) will either be a positive number or zero but never negative.

Real Numbers

Real numbers are an extensive set of numbers that include both rational and irrational numbers. They encompass just about every type of number you encounter in everyday math, making them versatile for solving equations and operations like taking square roots.

• Rational numbers are numbers that can be expressed as a fraction, such as \(\frac{1}{2}\), 3, or 0.75.
• Irrational numbers cannot be expressed exactly as fractions, like \(\sqrt{2}\) or \(\pi\).

Real numbers give us a full picture of the number line, stretching from negative infinity through zero to positive infinity.
Understanding that every real number squared results in a non-negative number helps explain why \(\sqrt{x}\) can't be negative within the real numbers.

Equation Solving

Equation solving is a fundamental skill in mathematics. It involves finding the unknown value that satisfies an equation. When solving equations like \(\sqrt{x} = -6\), we analyze whether a solution exists based on the properties of mathematical operations, such as square roots.
The process usually involves:

• Identifying and understanding the operation involved – here, square roots, which can't produce negative results.
• Substituting potential solutions into the equation and checking if both sides equal.

As in our exercise, substituting \(x = 36\) gives \(\sqrt{36} = 6\), which does not equal \(-6\). This confirms that there is no real solution for the equation \(\sqrt{x} = -6\), as the operation inherently limits its outcomes to non-negative numbers.