Solve The Equation: ( 2 X − 1 ) ( X − 3 ) = ( 2 X − 3 ) ( X − 1 (2x - 1)(x - 3) = (2x - 3)(x - 1 ( 2 X − 1 ) ( X − 3 ) = ( 2 X − 3 ) ( X − 1 ]10. Solve The Equation: X ( 2 X + 6 ) = 2 ( X 2 − 5 X(2x + 6) = 2(x^2 - 5 X ( 2 X + 6 ) = 2 ( X 2 − 5 ]12. Solve The Equation: ( 2 X + 1 ) ( X − 4 ) + ( X − 2 ) 2 = 13 X ( X + 2 (2x + 1)(x - 4) + (x - 2)^2 = 13x(x + 2 ( 2 X + 1 ) ( X − 4 ) + ( X − 2 ) 2 = 13 X ( X + 2 ]14. Solve The Equation: $x(x - 1) = 2(x - 1)(x +

Introduction

Equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving four different types of equations, each with its unique challenges and requirements. We will break down each equation step by step, providing a clear and concise explanation of the solution process.

Equation 1: $(2x - 1)(x - 3) = (2x - 3)(x - 1)$

Step 1: Expand the Left Side of the Equation

To solve this equation, we first need to expand the left side using the distributive property. This will give us:

$(2x - 1)(x - 3) = 2x^2 - 6x - x + 3$

Step 2: Simplify the Left Side of the Equation

Now, we can simplify the left side by combining like terms:

$2x^2 - 7x + 3$

Step 3: Expand the Right Side of the Equation

Next, we need to expand the right side of the equation using the distributive property:

$(2x - 3)(x - 1) = 2x^2 - 2x - 3x + 3$

Step 4: Simplify the Right Side of the Equation

Now, we can simplify the right side by combining like terms:

$2x^2 - 5x + 3$

Step 5: Set the Two Sides Equal to Each Other

Now that we have expanded and simplified both sides of the equation, we can set them equal to each other:

$2x^2 - 7x + 3 = 2x^2 - 5x + 3$

Step 6: Subtract $2x^2$ from Both Sides of the Equation

To eliminate the quadratic term, we can subtract $2x^2$ from both sides of the equation:

$-7x + 3 = -5x + 3$

Step 7: Subtract 3 from Both Sides of the Equation

Next, we can subtract 3 from both sides of the equation to isolate the variable term:

$-7x = -5x$

Step 8: Add $5x$ to Both Sides of the Equation

Finally, we can add $5x$ to both sides of the equation to solve for $x$:

$-2x = 0$

Step 9: Divide Both Sides of the Equation by -2

To solve for $x$, we can divide both sides of the equation by -2:

$x = 0$

Equation 2: $x(2x + 6) = 2(x^2 - 5)$

Step 1: Distribute the $x$ on the Left Side of the Equation

To solve this equation, we first need to distribute the $x$ on the left side using the distributive property:

$2x^2 + 6x = 2x^2 - 10$

Step 2: Subtract $2x^2$ from Both Sides of the Equation

Next, we can subtract $2x^2$ from both sides of the equation to eliminate the quadratic term:

$6x = -10$

Step 3: Divide Both Sides of the Equation by 6

Finally, we can divide both sides of the equation by 6 to solve for $x$:

$x = -\frac{10}{6}$

Step 4: Simplify the Fraction

We can simplify the fraction by dividing both the numerator and denominator by their greatest common divisor, which is 2:

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

Equation 3: $(2x + 1)(x - 4) + (x - 2)^2 = 13x(x + 2)$

Step 1: Expand the Left Side of the Equation

To solve this equation, we first need to expand the left side using the distributive property:

$(2x + 1)(x - 4) = 2x^2 - 8x + x - 4$

Step 2: Simplify the Left Side of the Equation

Now, we can simplify the left side by combining like terms:

$2x^2 - 7x - 4$

Step 3: Expand the Right Side of the Equation

Next, we need to expand the right side of the equation using the distributive property:

$13x(x + 2) = 13x^2 + 26x$

Step 4: Add $(x - 2)^2$ to the Left Side of the Equation

Now, we can add $(x - 2)^2$ to the left side of the equation:

$2x^2 - 7x - 4 + x^2 - 4x + 4 = 13x^2 + 26x$

Step 5: Simplify the Left Side of the Equation

Now, we can simplify the left side by combining like terms:

$3x^2 - 11x = 13x^2 + 26x$

Step 6: Subtract $13x^2$ from Both Sides of the Equation

To eliminate the quadratic term, we can subtract $13x^2$ from both sides of the equation:

$-10x^2 - 11x = 26x$

Step 7: Add $11x$ to Both Sides of the Equation

Next, we can add $11x$ to both sides of the equation to isolate the variable term:

$-10x^2 = 37x$

Step 8: Divide Both Sides of the Equation by -10

Finally, we can divide both sides of the equation by -10 to solve for $x$:

$x^2 = -\frac{37}{10}x$

Step 9: Add $\frac{37}{10}x^2$ to Both Sides of the Equation

To get the equation in standard form, we can add $\frac{37}{10}x^2$ to both sides of the equation:

$\frac{37}{10}x^2 + x^2 = 0$

Step 10: Combine Like Terms

Now, we can combine like terms:

$\frac{47}{10}x^2 = 0$

Step 11: Divide Both Sides of the Equation by $\frac{47}{10}$

Finally, we can divide both sides of the equation by $\frac{47}{10}$ to solve for $x$:

$x^2 = 0$

Step 12: Take the Square Root of Both Sides of the Equation

To solve for $x$, we can take the square root of both sides of the equation:

$x = 0$

Equation 4: $x(x - 1) = 2(x - 1)(x + 2)$

Step 1: Distribute the $x$ on the Left Side of the Equation

To solve this equation, we first need to distribute the $x$ on the left side using the distributive property:

$x^2 - x = 2x^2 + 4x - 2x$

Step 2: Simplify the Left Side of the Equation

Now, we can simplify the left side by combining like terms:

Q&A: Solving Equations

Q: What is the first step in solving an equation? A: The first step in solving an equation is to read and understand the equation. This includes identifying the variables, constants, and any mathematical operations involved.

Q: What is the distributive property, and how is it used in solving equations? A: The distributive property is a mathematical concept that allows us to expand expressions by multiplying each term inside the parentheses by the term outside the parentheses. This is used in solving equations to expand and simplify expressions.

Q: How do I know when to add or subtract the same value from both sides of an equation? A: You should add or subtract the same value from both sides of an equation when you want to isolate the variable term. This is done to eliminate the constant term and solve for the variable.

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 is 1, while a quadratic equation is an equation in which the highest power of the variable is 2.

Q: How do I solve a quadratic equation? A: To solve a quadratic equation, you can use the quadratic formula, which is:

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

where $a$, $b$, and $c$ are the coefficients of the quadratic equation.

Q: What is the quadratic formula, and how is it used in solving quadratic equations? A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is used to find the solutions to quadratic equations in the form of $ax^2 + bx + c = 0$.

Q: How do I know when to use the quadratic formula? A: You should use the quadratic formula when you are solving a quadratic equation and the equation cannot be factored.

Q: What is the difference between a system of linear equations and a system of quadratic equations? A: A system of linear equations is a set of two or more linear equations that are solved simultaneously, while a system of quadratic equations is a set of two or more quadratic equations that are solved simultaneously.

Q: How do I solve a system of linear equations? A: To solve a system of linear equations, you can use the method of substitution or the method of elimination.

Q: What is the method of substitution, and how is it used in solving systems of linear equations? A: The method of substitution is a mathematical technique that is used to solve systems of linear equations. It involves substituting the expression for one variable into the other equation to solve for the other variable.

Q: What is the method of elimination, and how is it used in solving systems of linear equations? A: The method of elimination is a mathematical technique that is used to solve systems of linear equations. It involves adding or subtracting the equations to eliminate one of the variables and solve for the other variable.

Q: How do I know when to use the method of substitution and when to use the method of elimination? A: You should use the method of substitution when one of the equations is already solved for one of the variables, and you should use the method of elimination when the equations are not easily solvable using substitution.

Conclusion

Solving equations is a crucial skill in mathematics, and it requires a deep understanding of mathematical concepts and techniques. By following the steps outlined in this article, you can solve a wide range of equations, from simple linear equations to complex quadratic equations. Remember to always read and understand the equation, identify the variables and constants, and use the appropriate mathematical techniques to solve the equation.