Solve for x Solve x^2=75
This problem is asking for the value of the variable \( x \) that satisfies the equation \( x^2 = 75 \). It requires finding the numerical value(s) of \( x \) when it is squared to produce the result of 75. This typically involves using algebraic methods to isolate \( x \) and finding its square root(s).
Solve$x^{2} = 75$
Answer
Solution:
Step 1:
Apply the square root to both sides to remove the square from x. $x = \pm \sqrt{75}$
Step 2:
Break down $\pm \sqrt{75}$ into simpler terms.
Step 2.1:
Express 75 as a product of its prime factors. $x = \pm \sqrt{25 \cdot 3}$
Step 2.1.1:
Identify 25 as a perfect square within 75. $x = \pm \sqrt{25 \cdot 3}$
Step 2.1.2:
Represent 25 as $5^2$. $x = \pm \sqrt{5^2 \cdot 3}$
Step 2.2:
Extract the square root of the perfect square. $x = \pm 5\sqrt{3}$
Step 3:
Determine both the positive and negative solutions.
Step 3.1:
Calculate the positive solution using the plus sign. $x = 5\sqrt{3}$
Step 3.2:
Calculate the negative solution using the minus sign. $x = -5\sqrt{3}$
Step 3.3:
Combine both solutions for the final answer. $x = 5\sqrt{3}, -5\sqrt{3}$
Step 4:
Present the solution in various formats.
Exact Form: $x = 5\sqrt{3}, -5\sqrt{3}$
Decimal Form: $x \approx 8.66025404, -8.66025404$
Knowledge Notes:
To solve the equation $x^2 = 75$, we follow a systematic approach:
Square Roots: To solve for $x$ when $x^2$ is given, we take the square root of both sides. Since squaring a number can result in either a positive or negative value, we consider both possibilities, leading to $\pm \sqrt{75}$.
Simplification: The square root of a number can often be simplified if the number under the radical is a product of a perfect square and another number. In this case, 75 can be factored into $25 \cdot 3$, where 25 is a perfect square.
Extraction of Perfect Squares: Since the square root of a perfect square is an integer, we can take $5^2$ out from under the radical, simplifying the expression to $5\sqrt{3}$.
Positive and Negative Solutions: Because we have a $\pm$ symbol, we must consider both the positive and negative square roots, resulting in two solutions: $5\sqrt{3}$ and $-5\sqrt{3}$.
Representation of Solutions: The final solutions can be expressed in exact form (with the square root symbol) or in decimal form, which requires a calculator for approximation.
Relevant mathematical concepts used in this problem include the properties of square roots, factoring, and simplification of radical expressions. Understanding these concepts is crucial for solving quadratic equations and working with radical expressions in algebra.
