Problem 23 $x^{2}=150$... [FREE SOLUTION]

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Elementary Algebra

Laura Bracken, Ed Miller

$Math Studyset 91影视 Explanations$ Math

1 Edition

Chapter 9: Problem 23

$x^{2}=150$

Short Answer

Expert verified

x = \pm 5\sqrt{6}

Step by step solution

- Understand the Equation

The given equation is a quadratic equation of the form $x^2 = 150$. This means we need to find the value(s) of $x$ that satisfy this equation.

- Take the Square Root of Both Sides

To isolate $x$, take the square root of both sides of the equation. Remember that taking the square root of both sides will give us two solutions: one positive and one negative. So, $x = \pm\sqrt{150}$.

- Simplify the Square Root

Simplify $\sqrt{150}$ by factoring out perfect squares. We know that $150 = 25 \times 6 = 5^2 \times 6$. Thus, $\sqrt{150} = \sqrt{5^2 \times 6} = 5\sqrt{6}$.

- Write the Final Solutions

The final solutions to the equation are $x = 5\sqrt{6}$ and $x = -5\sqrt{6}$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

quadratic equations

A quadratic equation is any equation that can be written in the form of $ax^2 + bx + c = 0$, where $a$, $b$, and $c$ are constants, and $x$ represents an unknown variable. However, in our specific exercise, we have a simpler form: $x^2 = 150$. This is still a quadratic equation because it has the $x^2$ term. Quadratic equations can often have two solutions. These solutions may be real or complex, depending on the term under the square root. In our case, we are dealing with real numbers.

square roots

The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 25 is 5 because $5 \times 5 = 25$. When you take the square root of both sides of an equation, it helps to isolate the variable. Remember that taking the square root can produce both positive and negative results. So, $x^2 = 150$ becomes $x = \pm\sqrt{150}$. Why do we get two results? Because $5 \sqrt{6} \times 5 \sqrt{6} = 150$ and $-5 \sqrt{6} \times -5 \sqrt{6} = 150$ as well.

simplifying radicals

Simplifying radicals means breaking down a complex square root into a simpler form. For example, $\sqrt{150}$ is not as simple as it could be. To simplify, we look for perfect square factors of 150. We notice that $25 \times 6 = 150$, and since 25 is a perfect square ($5^2$), we can rewrite:

\sqrt{150} = \sqrt{25 \times 6}
We know \sqrt{25} = 5, so our expression becomes 5\sqrt{6}

Thus, the simplified form of \sqrt{150} is 5\sqrt{6}. This method of simplifying helps in making further calculations easier and more manageable.

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91影视

\(x^{2}=150\)

Short Answer

Step by step solution

- Understand the Equation

- Take the Square Root of Both Sides

- Simplify the Square Root

- Write the Final Solutions

Key Concepts

quadratic equations

square roots

simplifying radicals

One App. One Place for Learning.

Most popular questions from this chapter

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