Problem 118
Question
Use any convenient method to solve the quadratic equation. $$25 x^{2}-1=0$$
Step-by-Step Solution
Verified Answer
The solutions to the equation are \(x = 0.2\) and \(x = -0.2\)
1Step 1: Isolate the x-squared term
Rewrite the equation as \(25x^2 = 1\). We are doing this to make it easier to solve for \(x\). This is achieved by adding 1 to both sides.
2Step 2: Solve for the x term
We now need to solve for \(x\). This is done by dividing both sides of the equation by 25, resulting in \(x^2 = 0.04\)
3Step 3: Solve for x
To solve for \(x\), take the square root of each side. Remember that taking the square root of any number always results in two possible answers: a positive and a negative one. Therefore, \(x = \sqrt{0.04}\) and \(x = -\sqrt{0.04}\). This can simplify to \(x = 0.2\) and \(x = -0.2\) respectively.
Key Concepts
Isolating VariablesSquare RootsQuadratic Formula
Isolating Variables
Isolating variables is a fundamental process in solving algebraic equations, including quadratic equations. It involves moving terms around in an equation such that one side of the equation contains the variable we are solving for, while the other side contains constants or numbers.
This is usually one of the first steps and is critical for simplifying the equation to a form where standard methods can easily be applied to find the variable's value. For instance, in the given equation, \(25x^2 - 1 = 0\), the initial step is to isolate the \(x^2\) term by adding 1 to both sides, resulting in \(25x^2 = 1\).
Next, to further isolate \(x\), divide both sides by 25, which simplifies the equation to \(x^2 = 0.04\). Now, \(x\) is set to be solved with additional methods such as taking square roots or applying the quadratic formula.
This is usually one of the first steps and is critical for simplifying the equation to a form where standard methods can easily be applied to find the variable's value. For instance, in the given equation, \(25x^2 - 1 = 0\), the initial step is to isolate the \(x^2\) term by adding 1 to both sides, resulting in \(25x^2 = 1\).
Next, to further isolate \(x\), divide both sides by 25, which simplifies the equation to \(x^2 = 0.04\). Now, \(x\) is set to be solved with additional methods such as taking square roots or applying the quadratic formula.
Square Roots
Solving a quadratic equation often brings us to the point of taking square roots, especially once the variable has been isolated and we're left with \(x^2\) equal to some number. When we take the square root of both sides, we must remember that we're dealing with two possible solutions: one positive and one negative.
For example, once \(x^2 = 0.04\) is achieved, taking the square root gives us \(x = \pm\sqrt{0.04}\). The reason for this is the fundamental nature of squares and square roots: if \(a^2 = b\), then \(a = \sqrt{b}\) or \(a = -\sqrt{b}\), because squaring either the positive or negative root of \(b\) will result in \(b\).
In our case, both \(0.2^2\) and \( (-0.2)^2\) yield 0.04, hence \(x = 0.2\) or \(x = -0.2\). Understanding the concept of square roots is key to solving many quadratic equations and cannot be understated.
For example, once \(x^2 = 0.04\) is achieved, taking the square root gives us \(x = \pm\sqrt{0.04}\). The reason for this is the fundamental nature of squares and square roots: if \(a^2 = b\), then \(a = \sqrt{b}\) or \(a = -\sqrt{b}\), because squaring either the positive or negative root of \(b\) will result in \(b\).
In our case, both \(0.2^2\) and \( (-0.2)^2\) yield 0.04, hence \(x = 0.2\) or \(x = -0.2\). Understanding the concept of square roots is key to solving many quadratic equations and cannot be understated.
Quadratic Formula
The quadratic formula is a powerful tool that provides a direct way to solve any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is \(x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\).
It encompasses the square root concept within it and derives the two possible solutions for \(x\) at once. This formula is derived from the process of completing the square and is universally applicable for all quadratic equations, irrespective of whether they are factorable or not. Using this formula can sometimes be quicker than other methods, especially when dealing with complex coefficients.
In the example of \(25x^2 - 1 = 0\), you could apply the quadratic formula with \(a=25\), \(b=0\), and \(c=-1\), but, as noted in the exercise, simpler methods like isolating \(x\) and taking square roots were sufficient. However, having a solid grasp of the quadratic formula is essential for more complex equations where isolating the variable is not as straightforward.
It encompasses the square root concept within it and derives the two possible solutions for \(x\) at once. This formula is derived from the process of completing the square and is universally applicable for all quadratic equations, irrespective of whether they are factorable or not. Using this formula can sometimes be quicker than other methods, especially when dealing with complex coefficients.
In the example of \(25x^2 - 1 = 0\), you could apply the quadratic formula with \(a=25\), \(b=0\), and \(c=-1\), but, as noted in the exercise, simpler methods like isolating \(x\) and taking square roots were sufficient. However, having a solid grasp of the quadratic formula is essential for more complex equations where isolating the variable is not as straightforward.
Other exercises in this chapter
Problem 117
Use any convenient method to solve the quadratic equation. $$4 x^{2}-17=0$$
View solution Problem 117
Determine whether the statement is true or false. Justify your answer. The graph of the function \(f(x)=x^{6}-x^{7}\) rises to the left and falls to the right.
View solution Problem 118
Determine whether the statement is true or false. Justify your answer. The graph of the function \(f(x)=2 x(x-1)^{2}(x+3)^{3}\) crosses the \(x\) -axis at \(x=1
View solution Problem 119
Use any convenient method to solve the quadratic equation. $$3 x^{2}-11 x-20=0$$
View solution