The First Step in Solving the Equation

Oct 13, 2025 by talynt.org 43 views

Understanding the Problem

When solving a quadratic equation, the first step is to isolate the variable $x$ by getting rid of any constants on the same side of the equation. In the given equation $x^2 - \frac{16}{25} = 0$ , the first step involves moving the constant term to the other side of the equation. This will allow us to set the equation equal to zero and prepare it for further simplification.

Isolating the Variable

To isolate the variable $x$ , we need to add $\frac{16}{25}$ to both sides of the equation. This will eliminate the constant term and leave us with a quadratic expression equal to zero. The equation becomes:

$x^2 = \frac{16}{25}$

Understanding the Right-Hand Side

The right-hand side of the equation is a fraction, $\frac{16}{25}$ . This fraction can be rewritten as a decimal or left as a fraction. However, for the purpose of solving the equation, it is more convenient to leave it as a fraction. The fraction $\frac{16}{25}$ can be thought of as a ratio of the square of a number to itself.

Understanding the Left-Hand Side

The left-hand side of the equation is a quadratic expression, $x^2$ . This expression represents the square of the variable $x$ . When we set the equation equal to zero, we are essentially looking for the values of $x$ that make the quadratic expression equal to the fraction on the right-hand side.

The First Step in Solving the Equation

The first step in solving the equation $x^2 - \frac{16}{25} = 0$ is to isolate the variable $x$ by moving the constant term to the other side of the equation. This is achieved by adding $\frac{16}{25}$ to both sides of the equation, resulting in:

$x^2 = \frac{16}{25}$

What is the Second Step in Solving the Equation?

Understanding the Second Step

The second step in solving the equation involves finding the square root of both sides of the equation. This will allow us to isolate the variable $x$ and find its possible values. When we take the square root of both sides of the equation, we must remember to consider both the positive and negative square roots.

Finding the Square Root

To find the square root of both sides of the equation, we can use the following property of square roots:

$\sqrt{a^2} = |a|$

This property states that the square root of a squared number is equal to the absolute value of the number. Applying this property to the equation, we get:

$x = \pm \sqrt{\frac{16}{25}}$

Simplifying the Square Root

The square root of $\frac{16}{25}$ can be simplified by finding the square root of the numerator and the denominator separately. The square root of 16 is 4, and the square root of 25 is 5. Therefore, the square root of $\frac{16}{25}$ is:

$\sqrt{\frac{16}{25}} = \frac{4}{5}$

Finding the Possible Values of $x$

Now that we have simplified the square root, we can find the possible values of $x$ by considering both the positive and negative square roots. The possible values of $x$ are:

$x = \pm \frac{4}{5}$

Conclusion

In conclusion, the first step in solving the equation $x^2 - \frac{16}{25} = 0$ is to isolate the variable $x$ by moving the constant term to the other side of the equation. The second step involves finding the square root of both sides of the equation and simplifying the square root to find the possible values of $x$ . The possible values of $x$ are $x = \pm \frac{4}{5}$ .

Final Answer

The final answer is: $\boxed{\frac{4}{5}}$

Q: What is the first step in solving the equation $x^2 - \frac{16}{25} = 0$ ?

A: The first step in solving the equation $x^2 - \frac{16}{25} = 0$ is to isolate the variable $x$ by moving the constant term to the other side of the equation. This is achieved by adding $\frac{16}{25}$ to both sides of the equation, resulting in:

$x^2 = \frac{16}{25}$

Q: Why do we need to isolate the variable $x$ ?

A: We need to isolate the variable $x$ to set the equation equal to zero and prepare it for further simplification. This will allow us to find the possible values of $x$ that make the quadratic expression equal to the fraction on the right-hand side.

Q: What is the second step in solving the equation $x^2 - \frac{16}{25} = 0$ ?

A: The second step in solving the equation involves finding the square root of both sides of the equation. This will allow us to isolate the variable $x$ and find its possible values. When we take the square root of both sides of the equation, we must remember to consider both the positive and negative square roots.

Q: How do we find the square root of both sides of the equation?

A: To find the square root of both sides of the equation, we can use the following property of square roots:

$\sqrt{a^2} = |a|$

This property states that the square root of a squared number is equal to the absolute value of the number. Applying this property to the equation, we get:

$x = \pm \sqrt{\frac{16}{25}}$

Q: How do we simplify the square root of $\frac{16}{25}$ ?

A: The square root of $\frac{16}{25}$ can be simplified by finding the square root of the numerator and the denominator separately. The square root of 16 is 4, and the square root of 25 is 5. Therefore, the square root of $\frac{16}{25}$ is:

$\sqrt{\frac{16}{25}} = \frac{4}{5}$

Q: What are the possible values of $x$ ?

A: The possible values of $x$ are:

$x = \pm \frac{4}{5}$

Q: Why do we consider both the positive and negative square roots?

A: We consider both the positive and negative square roots because the square root of a number can be both positive and negative. For example, the square root of 16 is both 4 and -4. Therefore, we must consider both the positive and negative square roots to find all possible values of $x$ .

Q: What is the final answer to the equation $x^2 - \frac{16}{25} = 0$ ?

A: The final answer to the equation $x^2 - \frac{16}{25} = 0$ is:

$x = \pm \frac{4}{5}$

Q: How do we check our answer?

A: To check our answer, we can plug the possible values of $x$ back into the original equation and see if they satisfy the equation. If they do, then our answer is correct.

Q: What if the equation has no solution?

A: If the equation has no solution, then the quadratic expression on the left-hand side of the equation is never equal to the fraction on the right-hand side. In this case, the equation has no real solution.

Q: What if the equation has multiple solutions?

A: If the equation has multiple solutions, then the quadratic expression on the left-hand side of the equation is equal to the fraction on the right-hand side for multiple values of $x$ . In this case, the equation has multiple real solutions.

Q: How do we find the solutions to a quadratic equation?

A: To find the solutions to a quadratic equation, we can use the quadratic formula:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

This formula gives us the possible values of $x$ that satisfy the quadratic equation.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that gives us the possible values of $x$ that satisfy a quadratic equation. The formula is:

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Q: How do we use the quadratic formula?

A: To use the quadratic formula, we need to plug in the values of $a$ , $b$ , and $c$ into the formula. We can then simplify the expression and find the possible values of $x$ .

Q: What are the values of $a$ , $b$ , and $c$ in the quadratic formula?

A: The values of $a$ , $b$ , and $c$ in the quadratic formula are the coefficients of the quadratic equation. In the equation $x^2 - \frac{16}{25} = 0$ , the values of $a$ , $b$ , and $c$ are:

$a = 1$ $b = 0$ $c = -\frac{16}{25}$

Q: How do we simplify the quadratic formula?

A: To simplify the quadratic formula, we can plug in the values of $a$ , $b$ , and $c$ into the formula and simplify the expression. We can then find the possible values of $x$ .

Q: What is the final answer to the quadratic formula?

A: The final answer to the quadratic formula is:

$x = \pm \frac{4}{5}$