Rewrite The Equation \[$\frac{x^3}{3}-\frac{1}{2x^2}+4=0\$\] In The Form \[$ax^n + Bx^{n-1} - 3 = 0\$\].Find The Value Of \[$a\$\], The Value Of \[$b\$\], And The Value Of \[$n\$\].
Introduction
In mathematics, equations are a fundamental concept that helps us understand and solve problems. One of the essential skills in algebra is rewriting equations in different forms to make them easier to solve. In this article, we will focus on rewriting the given equation {\frac{x3}{3}-\frac{1}{2x2}+4=0$}$ in the form {ax^n + bx^{n-1} - 3 = 0$}$. We will also find the values of {a$}$, {b$}$, and {n$}$.
Step 1: Move the Constant Term
The first step in rewriting the equation is to move the constant term to the right-hand side. This will help us isolate the terms with variables.
{\frac{x3}{3}-\frac{1}{2x2}+4=0$}$
Subtract 4 from both sides:
{\frac{x3}{3}-\frac{1}{2x2}=-4$}$
Step 2: Multiply Both Sides by 6x^2
To eliminate the fractions, we can multiply both sides of the equation by 6x^2. This will help us get rid of the denominators.
{\frac{x3}{3}-\frac{1}{2x2}=-4$}$
Multiply both sides by 6x^2:
${6x^5-3=-24x^2\$}
Step 3: Rearrange the Terms
Now, we can rearrange the terms to get the equation in the desired form.
${6x^5-3=-24x^2\$}
Add 24x^2 to both sides:
${6x^5+24x^2-3=0\$}
Step 4: Identify the Values of a, b, and n
Now that we have the equation in the desired form, we can identify the values of {a$}$, {b$}$, and {n$}$.
${6x^5+24x^2-3=0\$}
Comparing this equation with the desired form {ax^n + bx^{n-1} - 3 = 0$}$, we can see that:
- {a=6$}$
- {b=24$}$
- {n=5$}$
Conclusion
In this article, we have rewritten the given equation {\frac{x3}{3}-\frac{1}{2x2}+4=0$}$ in the form {ax^n + bx^{n-1} - 3 = 0$}$. We have also found the values of {a$}$, {b$}$, and {n$}$. The values are {a=6$}$, {b=24$}$, and {n=5$}$.
Tips and Tricks
- When rewriting equations, it's essential to move the constant term to the right-hand side first.
- Multiplying both sides of the equation by a common factor can help eliminate fractions.
- Rearranging the terms can help get the equation in the desired form.
Frequently Asked Questions
- Q: How do I rewrite an equation in a different form? A: To rewrite an equation, you can move the constant term to the right-hand side, multiply both sides by a common factor, and rearrange the terms.
- Q: What is the value of {a$}$ in the equation {ax^n + bx^n-1} - 3 = 0$}$? A$ is the coefficient of the highest power of x in the equation.
- Q: How do I find the value of {n$}$ in the equation {ax^n + bx^n-1} - 3 = 0$}$?
A$ is the power of x in the highest power term of the equation.
Frequently Asked Questions: Rewriting Equations =====================================================
Q: What is the purpose of rewriting equations?
A: Rewriting equations is an essential skill in mathematics that helps us simplify and solve problems. By rewriting equations in different forms, we can make them easier to understand and work with.
Q: How do I rewrite an equation in a different form?
A: To rewrite an equation, you can follow these steps:
- Move the constant term to the right-hand side.
- Multiply both sides of the equation by a common factor to eliminate fractions.
- Rearrange the terms to get the equation in the desired form.
Q: What is the value of a in the equation ax^n + bx^(n-1) - 3 = 0?
A: The value of a is the coefficient of the highest power of x in the equation.
Q: How do I find the value of n in the equation ax^n + bx^(n-1) - 3 = 0?
A: The value of n is the power of x in the highest power term of the equation.
Q: What is the difference between a and b in the equation ax^n + bx^(n-1) - 3 = 0?
A: The values of a and b are coefficients of the terms in the equation. The value of a is the coefficient of the highest power of x, while the value of b is the coefficient of the term with the next highest power of x.
Q: Can I rewrite an equation with a negative exponent?
A: Yes, you can rewrite an equation with a negative exponent. To do this, you can use the rule that a^(-n) = 1/a^n.
Q: How do I rewrite an equation with a fraction?
A: To rewrite an equation with a fraction, you can multiply both sides of the equation by the denominator of the fraction. This will help you eliminate the fraction.
Q: What is the value of x in the equation x^2 + 4x + 4 = 0?
A: To find the value of x, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. In this case, a = 1, b = 4, and c = 4.
Q: Can I rewrite an equation with a radical?
A: Yes, you can rewrite an equation with a radical. To do this, you can use the rule that √(a^2) = a.
Q: How do I rewrite an equation with a trigonometric function?
A: To rewrite an equation with a trigonometric function, you can use the trigonometric identities to simplify the equation.
Q: What is the value of a in the equation 2x^2 + 5x + 3 = 0?
A: To find the value of a, you can compare the equation with the general form ax^2 + bx + c = 0. In this case, a = 2.
Q: Can I rewrite an equation with a complex number?
A: Yes, you can rewrite an equation with a complex number. To do this, you can use the rules for working with complex numbers.
Q: How do I rewrite an equation with a matrix?
A: To rewrite an equation with a matrix, you can use the rules for working with matrices.
Conclusion
Rewriting equations is an essential skill in mathematics that helps us simplify and solve problems. By following the steps outlined in this article, you can rewrite equations in different forms and make them easier to understand and work with.