1. Solve The Following Equations: A. $\log_4(x^2 - 3x) = 1$ B. Solve $3^{2x} - 4(3^x) + 3 = 0$2. Solve The Equation $2a^2 + 4a = 3$, Giving Your Answer To 3 Significant Figures.3. Factor Completely: $3 + 5xy -

Introduction

Mathematical equations are an essential part of mathematics, and solving them is a crucial skill that every student and professional should possess. In this article, we will focus on solving three different types of equations: logarithmic, quadratic, and polynomial equations. We will provide step-by-step solutions to each equation, and we will also discuss the importance of solving mathematical equations.

Solving Logarithmic Equations

Logarithmic equations are equations that involve logarithmic functions. A logarithmic function is a function that takes a number as input and returns a value that represents the power to which a base number must be raised to produce the input number. In this section, we will solve the following logarithmic equation:

a. $\log_4(x^2 - 3x) = 1$

To solve this equation, we need to use the definition of a logarithmic function. The definition of a logarithmic function is:

$$\log_b(x) = y \iff b^y = x$$

Using this definition, we can rewrite the equation as:

$$4^1 = x^2 - 3x$$

Simplifying the equation, we get:

$$4 = x^2 - 3x$$

Rearranging the equation, we get:

$$x^2 - 3x - 4 = 0$$

This is a quadratic equation, and we can solve it using the quadratic formula:

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

In this case, $a = 1$, $b = -3$, and $c = -4$. Plugging these values into the formula, we get:

$$x = \frac{-(-3) \pm \sqrt{(-3)^2 - 4(1)(-4)}}{2(1)}$$

Simplifying the equation, we get:

$$x = \frac{3 \pm \sqrt{9 + 16}}{2}$$

$$x = \frac{3 \pm \sqrt{25}}{2}$$

$$x = \frac{3 \pm 5}{2}$$

Therefore, the solutions to the equation are:

$$x = \frac{3 + 5}{2} = 4$$

$$x = \frac{3 - 5}{2} = -1$$

b. Solve $3^{2x} - 4(3^x) + 3 = 0$

To solve this equation, we can use the substitution method. Let's substitute $y = 3^x$ into the equation:

$$y^2 - 4y + 3 = 0$$

This is a quadratic equation, and we can solve it using the quadratic formula:

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

In this case, $a = 1$, $b = -4$, and $c = 3$. Plugging these values into the formula, we get:

$$y = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)}$$

Simplifying the equation, we get:

$$y = \frac{4 \pm \sqrt{16 - 12}}{2}$$

$$y = \frac{4 \pm \sqrt{4}}{2}$$

$$y = \frac{4 \pm 2}{2}$$

Therefore, the solutions to the equation are:

$$y = \frac{4 + 2}{2} = 3$$

$$y = \frac{4 - 2}{2} = 1$$

Now, we can substitute back $y = 3^x$ into the equation:

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) = -3$$

$$3^{2x} = 4(3^x) - 3$$

$$3^{2x} = 3(3^x) + 3^x - 3$$

$$3^{2x} = 3(3^x)(1 + \frac{1}{3}) - 3$$

$$3^{2x} = 3(3^x)(\frac{4}{3}) - 3$$

$$3^{2x} = 4(3^x) - 3$$

$$3^{2x} = 4(3^x) - 3$$

$$3^{2x} - 4(3^x) = -3$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

$$3^{2x} - 4(3^x) + 3 = 0$$

3^{2x} - 4(3^x) + 3<br/> **Solving Mathematical Equations: A Comprehensive Guide** **Q&A Section** **Q: What is the difference between a logarithmic equation and a quadratic equation?** A: A logarithmic equation is an equation that involves a logarithmic function, which is a function that takes a number as input and returns a value that represents the power to which a base number must be raised to produce the input number. A quadratic equation, on the other hand, is an equation that involves a quadratic function, which is a function of the form $f(x) = ax^2 + bx + c$. **Q: How do I solve a logarithmic equation?** A: To solve a logarithmic equation, you need to use the definition of a logarithmic function, which is: $\log_b(x) = y \iff b^y = x

Using this definition, you can rewrite the equation as an exponential equation and then solve for the variable.

Q: What is the quadratic formula?

A: The quadratic formula is a formula that is used to solve quadratic equations. It is given by:

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

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 and then simplify the expression.

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

A: A quadratic equation is an equation that involves a quadratic function, which is a function of the form $f(x) = ax^2 + bx + c$. A polynomial equation, on the other hand, is an equation that involves a polynomial function, which is a function of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.

Q: How do I factor a polynomial equation?

A: To factor a polynomial equation, you need to find the greatest common factor (GCF) of the terms and then factor out the GCF.

Q: What is the greatest common factor (GCF)?

A: The greatest common factor (GCF) is the largest factor that divides all the terms of a polynomial equation.

Q: How do I find the GCF of a polynomial equation?

A: To find the GCF of a polynomial equation, you need to list all the factors of each term and then find the greatest common factor.

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

A: A linear equation is an equation that involves a linear function, which is a function of the form $f(x) = ax + b$. A quadratic equation, on the other hand, is an equation that involves a quadratic function, which is a function of the form $f(x) = ax^2 + bx + c$.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

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

A: A rational equation is an equation that involves a rational function, which is a function of the form $f(x) = \frac{p(x)}{q(x)}$. A polynomial equation, on the other hand, is an equation that involves a polynomial function, which is a function of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.

Q: How do I solve a rational equation?

A: To solve a rational equation, you need to find the least common multiple (LCM) of the denominators and then multiply both sides of the equation by the LCM.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that is a common multiple of two or more numbers.

Q: How do I find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, you need to list all the multiples of each number and then find the smallest multiple that is common to all the numbers.

Q: What is the difference between a system of equations and a single equation?

A: A system of equations is a set of two or more equations that are solved simultaneously. A single equation, on the other hand, is a single equation that is solved independently.

Q: How do I solve a system of equations?

A: To solve a system of equations, you need to use the method of substitution or elimination to find the values of the variables.

Q: What is the method of substitution?

A: The method of substitution is a method of solving a system of equations by substituting the expression for one variable into the other equation.

Q: What is the method of elimination?

A: The method of elimination is a method of solving a system of equations by adding or subtracting the equations to eliminate one of the variables.

Q: How do I use the method of substitution to solve a system of equations?

A: To use the method of substitution to solve a system of equations, you need to substitute the expression for one variable into the other equation and then solve for the other variable.

Q: How do I use the method of elimination to solve a system of equations?

A: To use the method of elimination to solve a system of equations, you need to add or subtract the equations to eliminate one of the variables and then solve for the other variable.

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

A: A linear inequality is an inequality that involves a linear function, which is a function of the form $f(x) = ax + b$. A quadratic inequality, on the other hand, is an inequality that involves a quadratic function, which is a function of the form $f(x) = ax^2 + bx + c$.

Q: How do I solve a linear inequality?

A: To solve a linear inequality, you need to isolate the variable by adding, subtracting, multiplying, or dividing both sides of the inequality by the same value.

Q: How do I solve a quadratic inequality?

A: To solve a quadratic inequality, you need to find the roots of the quadratic equation and then use the roots to determine the intervals where the inequality is true.

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

A: A rational inequality is an inequality that involves a rational function, which is a function of the form $f(x) = \frac{p(x)}{q(x)}$. A polynomial inequality, on the other hand, is an inequality that involves a polynomial function, which is a function of the form $f(x) = a_nx^n + a_{n-1}x^{n-1} + \ldots + a_1x + a_0$.

Q: How do I solve a rational inequality?

A: To solve a rational inequality, you need to find the least common multiple (LCM) of the denominators and then multiply both sides of the inequality by the LCM.

Q: How do I solve a polynomial inequality?

A: To solve a polynomial inequality, you need to find the roots of the polynomial equation and then use the roots to determine the intervals where the inequality is true.

Q: What is the difference between a system of inequalities and a single inequality?

A: A system of inequalities is a set of two or more inequalities that are solved simultaneously. A single inequality, on the other hand, is a single inequality that is solved independently.

Q: How do I solve a system of inequalities?

A: To solve a system of inequalities, you need to use the method of substitution or elimination to find the values of the variables.

Q: What is the method of substitution for solving a system of inequalities?

A: The method of substitution is a method of solving a system of inequalities by substituting the expression for one variable into the other inequality.

Q: What is the method of elimination for solving a system of inequalities?

A: The method of elimination is a method of solving a system of inequalities by adding or subtracting the inequalities to eliminate one of the variables.

Q: How do I use the method of substitution to solve a system of inequalities?

A: To use the method of substitution to solve a system of inequalities, you need to substitute the expression for one variable into the other inequality and then solve for the other variable.

Q: How do I use the method of elimination to solve a system of inequalities?

A: To use the method of elimination to solve a system of inequalities, you need to add or subtract the inequalities to eliminate one of the variables and then solve for the other variable.

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

A: A linear equation is an equation that involves a linear function, which is a function of the form $f(x) = ax + b$. A nonlinear equation, on the other hand, is an equation that involves a nonlinear function, which is a function of the form $f(x) = ax^2 + bx + c$ or $f(x) = ax^3 + bx^2 + cx + d$.

Q: How do I solve a linear equation?

A: To solve a linear equation, you need to isolate the variable by adding, subtracting, multiplying, or dividing