Problem 78
Question
Solve each equation in Exercises \(73-98\) by the method of your choice. \(2 x^{2}=250\)
Step-by-Step Solution
Verified Answer
The solutions to the equation \(2 x^{2}=250\) are \(x = \sqrt{125}\) and \(x = -\sqrt{125}\)
1Step 1: Isolate the terms
Divide both sides of the equation by 2 to isolate the term with the variable \(x\).\nHence, the equation becomes \(x^{2}=125\).
2Step 2: Solve for x
Take the square root of both sides of the equation to solve for \(x\). This will result in two values for \(x\), namely \(x = \sqrt{125}\) and \(x = - \sqrt{125}\).
3Step 3: Simplify
Simplify the square root of both roots to get the final values of \(x\). Since the square root of 125 is not a perfect square, we will leave it as a root value. Hence, \(x = \sqrt{125}\) or \(x = -\sqrt{125}\)
Key Concepts
Isolate VariablesSquare Root MethodAlgebraic Solutions
Isolate Variables
Isolating variables is a critical first step in solving algebraic equations, particularly quadratic equations. It involves rearranging the equation so that one side contains only variables and the other side contains only numbers. This creates a clearer path to finding the solution.
For example, to solve the quadratic equation in the given exercise, one must first isolate the squared term, which is achieved by dividing both sides by the constant that multiplies the variable squate, in this case, 2. After performing the operation, we get a much simpler equation: \(x^2 = 125\). This technique simplifies complex equations and lays the groundwork for further operations such as taking the square root, which is the next step towards finding the solution.
Understanding this step is essential as it often simplifies the equation, making the subsequent steps easier. Remember, for every operation you perform, it must be done to both sides of the equation to maintain its balance.
For example, to solve the quadratic equation in the given exercise, one must first isolate the squared term, which is achieved by dividing both sides by the constant that multiplies the variable squate, in this case, 2. After performing the operation, we get a much simpler equation: \(x^2 = 125\). This technique simplifies complex equations and lays the groundwork for further operations such as taking the square root, which is the next step towards finding the solution.
Understanding this step is essential as it often simplifies the equation, making the subsequent steps easier. Remember, for every operation you perform, it must be done to both sides of the equation to maintain its balance.
Square Root Method
The square root method is a powerful way to solve quadratic equations after one has successfully isolated the variable term. When a quadratic equation is formulated as \(x^2 = c\), where \(c\) is a constant, taking the square root of both sides helps find the value of \(x\).
However, an important aspect of this method is to remember that every positive number has two square roots: one positive and one negative. This is why, in the given exercise solution, we find two potential values for \(x\): \(x = \sqrt{125}\) and \(x = -\sqrt{125}\). The reason we take both the positive and negative roots is that squaring either the negative or the positive version of the value will result in the positive constant we started with. Even though \(\sqrt{125}\) is not a perfect square, simplifying it is not always necessary for finding a viable solution.
However, an important aspect of this method is to remember that every positive number has two square roots: one positive and one negative. This is why, in the given exercise solution, we find two potential values for \(x\): \(x = \sqrt{125}\) and \(x = -\sqrt{125}\). The reason we take both the positive and negative roots is that squaring either the negative or the positive version of the value will result in the positive constant we started with. Even though \(\sqrt{125}\) is not a perfect square, simplifying it is not always necessary for finding a viable solution.
Algebraic Solutions
Algebraic solutions encompass various methods used to solve algebraic equations with one or more variables. For quadratic equations, in particular, there are multiple techniques to find the solution, such as factoring, completing the square, or using the quadratic formula.
The choice of method often depends on the structure of the equation and the preference of the person solving it. In the step by step solution for the exercise, we used the square root method due to the equation's ready-to-square-root form after isolating \(x^2\).
It's important to recognize that whichever algebraic solution method is chosen, a fundamental understanding of manipulating equations and performing arithmetic operations will be crucial. Thus, cementing these basic skills will make approaching quadratic equations, and algebra as a whole, far more manageable.
The choice of method often depends on the structure of the equation and the preference of the person solving it. In the step by step solution for the exercise, we used the square root method due to the equation's ready-to-square-root form after isolating \(x^2\).
It's important to recognize that whichever algebraic solution method is chosen, a fundamental understanding of manipulating equations and performing arithmetic operations will be crucial. Thus, cementing these basic skills will make approaching quadratic equations, and algebra as a whole, far more manageable.
Other exercises in this chapter
Problem 77
Solve each inequality by first rewriting each one as an equivalent inequality without absolute value bars. Graph the solution set on a number line. Express the
View solution Problem 77
What is the solution set of an equation?
View solution Problem 78
Solve each equation by the method of your choice. $$ \left(x^{2}-1\right)^{2}-2\left(x^{2}-1\right)=3 $$
View solution Problem 78
We discussed formulas in this section after we considered procedures for solving linear equations. Doesn't working -with a formula simply mean substituting give
View solution