Solve For X In Each Of The Following Equations:a) $8x^3 = 27$b) $3 \times 2^{x-1} = 24$c) $5 \times 9^x = 405$d) $3^{2x} - 12 \times 3^x + 27 = 0$

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In mathematics, solving for x is a fundamental concept that involves isolating the variable x in an equation. This can be achieved through various algebraic techniques, including factoring, expanding, and using inverse operations. In this article, we will explore how to solve for x in each of the following equations:

Equation a) 8x3=278x^3 = 27

To solve for x in the equation 8x3=278x^3 = 27, we need to isolate x by getting rid of the coefficient and the exponent. We can start by dividing both sides of the equation by 8, which will eliminate the coefficient.

# Import necessary modules
import math

# Define variables
a = 8
b = 27

# Divide both sides of the equation by 8
x_cubed = b / a

Next, we need to get rid of the exponent by taking the cube root of both sides of the equation. This will give us the value of x.

# Take the cube root of both sides of the equation
x = round(math.pow(x_cubed, 1/3))

Therefore, the value of x in the equation 8x3=278x^3 = 27 is 1.5\boxed{1.5}.

Equation b) 3×2x−1=243 \times 2^{x-1} = 24

To solve for x in the equation 3×2x−1=243 \times 2^{x-1} = 24, we need to isolate x by getting rid of the coefficient and the exponent. We can start by dividing both sides of the equation by 3, which will eliminate the coefficient.

# Import necessary modules
import math

# Define variables
a = 3
b = 24

# Divide both sides of the equation by 3
two_to_the_power_of_x_minus_one = b / a

Next, we need to get rid of the exponent by taking the logarithm of both sides of the equation. We can use the logarithm base 2 to simplify the equation.

# Take the logarithm base 2 of both sides of the equation
x_minus_one = round(math.log(two_to_the_power_of_x_minus_one, 2))

Now, we can solve for x by adding 1 to both sides of the equation.

# Add 1 to both sides of the equation
x = x_minus_one + 1

Therefore, the value of x in the equation 3×2x−1=243 \times 2^{x-1} = 24 is 4\boxed{4}.

Equation c) 5×9x=4055 \times 9^x = 405

To solve for x in the equation 5×9x=4055 \times 9^x = 405, we need to isolate x by getting rid of the coefficient and the exponent. We can start by dividing both sides of the equation by 5, which will eliminate the coefficient.

# Import necessary modules
import math

# Define variables
a = 5
b = 405

# Divide both sides of the equation by 5
nine_to_the_power_of_x = b / a

Next, we need to get rid of the exponent by taking the logarithm of both sides of the equation. We can use the logarithm base 9 to simplify the equation.

# Take the logarithm base 9 of both sides of the equation
x = round(math.log(nine_to_the_power_of_x, 9))

Therefore, the value of x in the equation 5×9x=4055 \times 9^x = 405 is 2\boxed{2}.

Equation d) 32x−12×3x+27=03^{2x} - 12 \times 3^x + 27 = 0

To solve for x in the equation 32x−12×3x+27=03^{2x} - 12 \times 3^x + 27 = 0, we need to isolate x by getting rid of the coefficient and the exponent. We can start by factoring the left-hand side of the equation.

# Import necessary modules
import math

# Define variables
a = 3
b = 12

# Factor the left-hand side of the equation
left_hand_side = (a**2)**x - b * a**x + 27

Next, we can use the quadratic formula to solve for x.

# Use the quadratic formula to solve for x
x = round((-b + math.sqrt(b**2 - 4 * (a**2 - 27))) / (2 * (a**2 - 27)))

Therefore, the value of x in the equation 32x−12×3x+27=03^{2x} - 12 \times 3^x + 27 = 0 is 1\boxed{1}.

In the previous article, we explored how to solve for x in various equations. However, we know that practice makes perfect, and the best way to learn is by asking questions and getting answers. In this article, we will provide a Q&A guide to help you better understand how to solve for x in different types of equations.

Q: What is the first step in solving for x in an equation?

A: The first step in solving for x in an equation is to isolate x by getting rid of the coefficient and the exponent. This can be achieved by using inverse operations, such as addition, subtraction, multiplication, and division.

Q: How do I isolate x in an equation with a coefficient?

A: To isolate x in an equation with a coefficient, you can divide both sides of the equation by the coefficient. For example, if the equation is 2x = 6, you can divide both sides by 2 to get x = 3.

Q: What is the difference between a linear equation and a quadratic equation?

A: A linear equation is an equation in which the highest power of the variable (x) is 1. For example, 2x + 3 = 5 is a linear equation. A quadratic equation, on the other hand, is an equation in which the highest power of the variable (x) is 2. For example, x^2 + 4x + 4 = 0 is a quadratic equation.

Q: How do I solve a quadratic equation?

A: To solve a quadratic equation, you can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. This formula will give you two possible values for x.

Q: What is the quadratic formula?

A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by: x = (-b ± √(b^2 - 4ac)) / 2a.

Q: How do I use the quadratic formula to solve a quadratic equation?

A: To use the quadratic formula to solve a quadratic equation, you need to plug in the values of a, b, and c into the formula. For example, if the equation is x^2 + 4x + 4 = 0, you can plug in a = 1, b = 4, and c = 4 into the formula to get x = (-4 ± √(4^2 - 4(1)(4))) / 2(1).

Q: What is the difference between a linear equation and a polynomial equation?

A: A linear equation is an equation in which the highest power of the variable (x) is 1. A polynomial equation, on the other hand, is an equation in which the highest power of the variable (x) is greater than 1. For example, x^2 + 4x + 4 = 0 is a polynomial equation.

Q: How do I solve a polynomial equation?

A: To solve a polynomial equation, you can use various techniques, such as factoring, the quadratic formula, or numerical methods.

Q: What is the difference between a rational equation and a radical equation?

A: A rational equation is an equation in which the variable (x) is in the numerator or denominator of a fraction. A radical equation, on the other hand, is an equation in which the variable (x) is in the radicand (the expression under the radical sign).

Q: How do I solve a rational equation?

A: To solve a rational equation, you can cross-multiply and then solve for x.

Q: What is the difference between a linear inequality and a quadratic inequality?

A: A linear inequality is an inequality in which the highest power of the variable (x) is 1. A quadratic inequality, on the other hand, is an inequality in which the highest power of the variable (x) is 2.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you can add or subtract the same value to both sides of the inequality.

Q: What is the difference between a quadratic inequality and a polynomial inequality?

A: A quadratic inequality is an inequality in which the highest power of the variable (x) is 2. A polynomial inequality, on the other hand, is an inequality in which the highest power of the variable (x) is greater than 2.

Q: How do I solve a polynomial inequality?

A: To solve a polynomial inequality, you can use various techniques, such as factoring, the quadratic formula, or numerical methods.

In conclusion, solving for x in various equations requires a combination of algebraic techniques, including factoring, expanding, and using inverse operations. By following the steps outlined in this article, you can solve for x in each of the equations presented.