Solve The Following Equation For { X$} : : : {(x+2)\left(x-\frac{4}{2}\right)-(3x-1)\left(x+\frac{2}{3}\right)=1-2x\}

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Introduction

In this article, we will delve into solving a complex equation involving multiple variables and operations. The given equation is a quadratic equation that requires careful expansion and simplification to isolate the variable x. We will break down the solution step by step, using algebraic techniques to simplify the equation and ultimately solve for x.

The Given Equation

The given equation is:

(x+2)(x−42)−(3x−1)(x+23)=1−2x{(x+2)\left(x-\frac{4}{2}\right)-(3x-1)\left(x+\frac{2}{3}\right)=1-2x}

Step 1: Simplify the Equation

To simplify the equation, we will start by evaluating the expressions inside the parentheses.

(x+2)(x−42)=(x+2)(x−2){(x+2)\left(x-\frac{4}{2}\right) = (x+2)(x-2)} (3x−1)(x+23)=(3x−1)(3x+23){(3x-1)\left(x+\frac{2}{3}\right) = (3x-1)\left(\frac{3x+2}{3}\right)}

Step 2: Expand the Expressions

Next, we will expand the expressions using the distributive property.

(x+2)(x−2)=x2−2x+2x−4{(x+2)(x-2) = x^2 - 2x + 2x - 4} (3x−1)(3x+23)=13(9x2+6x−3x−2){(3x-1)\left(\frac{3x+2}{3}\right) = \frac{1}{3}(9x^2 + 6x - 3x - 2)}

Step 3: Simplify the Expanded Expressions

Now, we will simplify the expanded expressions.

x2−2x+2x−4=x2−4{x^2 - 2x + 2x - 4 = x^2 - 4} 13(9x2+6x−3x−2)=13(9x2+3x−2){\frac{1}{3}(9x^2 + 6x - 3x - 2) = \frac{1}{3}(9x^2 + 3x - 2)}

Step 4: Rewrite the Equation

Substituting the simplified expressions back into the original equation, we get:

x2−4−13(9x2+3x−2)=1−2x{x^2 - 4 - \frac{1}{3}(9x^2 + 3x - 2) = 1 - 2x}

Step 5: Distribute the Negative Sign

Distributing the negative sign to the terms inside the parentheses, we get:

x2−4−93x2−33x+23=1−2x{x^2 - 4 - \frac{9}{3}x^2 - \frac{3}{3}x + \frac{2}{3} = 1 - 2x}

Step 6: Combine Like Terms

Combining like terms, we get:

−83x2−13x−103=1−2x{-\frac{8}{3}x^2 - \frac{1}{3}x - \frac{10}{3} = 1 - 2x}

Step 7: Move All Terms to One Side

Moving all terms to one side of the equation, we get:

−83x2−13x−103−1+2x=0{-\frac{8}{3}x^2 - \frac{1}{3}x - \frac{10}{3} - 1 + 2x = 0}

Step 8: Simplify the Equation

Simplifying the equation, we get:

−83x2−13x+2x−133=0{-\frac{8}{3}x^2 - \frac{1}{3}x + 2x - \frac{13}{3} = 0}

Step 9: Combine Like Terms

Combining like terms, we get:

−83x2+53x−133=0{-\frac{8}{3}x^2 + \frac{5}{3}x - \frac{13}{3} = 0}

Step 10: Multiply Both Sides by -3

Multiplying both sides of the equation by -3, we get:

8x2−5x+13=0{8x^2 - 5x + 13 = 0}

Step 11: Solve the Quadratic Equation

To solve the quadratic equation, we can use the quadratic formula:

x=−b±b2−4ac2a{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}

In this case, a = 8, b = -5, and c = 13.

Step 12: Plug in the Values

Plugging in the values into the quadratic formula, we get:

x=−(−5)±(−5)2−4(8)(13)2(8){x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(8)(13)}}{2(8)}}

Step 13: Simplify the Expression

Simplifying the expression, we get:

x=5±25−41616{x = \frac{5 \pm \sqrt{25 - 416}}{16}}

Step 14: Simplify the Expression Further

Simplifying the expression further, we get:

x=5±−39116{x = \frac{5 \pm \sqrt{-391}}{16}}

Step 15: Simplify the Square Root

Simplifying the square root, we get:

x=5±i39116{x = \frac{5 \pm i\sqrt{391}}{16}}

Step 16: Simplify the Complex Number

Simplifying the complex number, we get:

x=516±i39116{x = \frac{5}{16} \pm \frac{i\sqrt{391}}{16}}

The final answer is 516±i39116\boxed{\frac{5}{16} \pm \frac{i\sqrt{391}}{16}}.

Introduction

In our previous article, we delved into solving a complex equation involving multiple variables and operations. The given equation was a quadratic equation that required careful expansion and simplification to isolate the variable x. In this article, we will provide a Q&A section to help clarify any doubts and provide additional insights into the solution.

Q: What is the given equation?

A: The given equation is:

(x+2)(x−42)−(3x−1)(x+23)=1−2x{(x+2)\left(x-\frac{4}{2}\right)-(3x-1)\left(x+\frac{2}{3}\right)=1-2x}

Q: How do I simplify the equation?

A: To simplify the equation, we need to evaluate the expressions inside the parentheses and then expand the resulting expressions using the distributive property.

Q: What is the first step in simplifying the equation?

A: The first step in simplifying the equation is to evaluate the expressions inside the parentheses.

Q: How do I evaluate the expressions inside the parentheses?

A: To evaluate the expressions inside the parentheses, we need to multiply the terms inside each set of parentheses.

Q: What is the result of evaluating the expressions inside the parentheses?

A: The result of evaluating the expressions inside the parentheses is:

(x+2)(x−2)=x2−2x+2x−4{(x+2)(x-2) = x^2 - 2x + 2x - 4} (3x−1)(3x+23)=13(9x2+6x−3x−2){(3x-1)\left(\frac{3x+2}{3}\right) = \frac{1}{3}(9x^2 + 6x - 3x - 2)}

Q: How do I simplify the expanded expressions?

A: To simplify the expanded expressions, we need to combine like terms.

Q: What is the result of simplifying the expanded expressions?

A: The result of simplifying the expanded expressions is:

x2−4{x^2 - 4} 13(9x2+3x−2){\frac{1}{3}(9x^2 + 3x - 2)}

Q: How do I rewrite the equation?

A: To rewrite the equation, we need to substitute the simplified expressions back into the original equation.

Q: What is the result of rewriting the equation?

A: The result of rewriting the equation is:

x2−4−13(9x2+3x−2)=1−2x{x^2 - 4 - \frac{1}{3}(9x^2 + 3x - 2) = 1 - 2x}

Q: How do I distribute the negative sign?

A: To distribute the negative sign, we need to multiply the terms inside the parentheses by -1.

Q: What is the result of distributing the negative sign?

A: The result of distributing the negative sign is:

x2−4−93x2−33x+23=1−2x{x^2 - 4 - \frac{9}{3}x^2 - \frac{3}{3}x + \frac{2}{3} = 1 - 2x}

Q: How do I combine like terms?

A: To combine like terms, we need to add or subtract the coefficients of the same variables.

Q: What is the result of combining like terms?

A: The result of combining like terms is:

−83x2−13x−103=1−2x{-\frac{8}{3}x^2 - \frac{1}{3}x - \frac{10}{3} = 1 - 2x}

Q: How do I move all terms to one side?

A: To move all terms to one side, we need to add or subtract the same value to both sides of the equation.

Q: What is the result of moving all terms to one side?

A: The result of moving all terms to one side is:

−83x2−13x+2x−133=0{-\frac{8}{3}x^2 - \frac{1}{3}x + 2x - \frac{13}{3} = 0}

Q: How do I simplify the equation further?

A: To simplify the equation further, we need to combine like terms.

Q: What is the result of simplifying the equation further?

A: The result of simplifying the equation further is:

−83x2+53x−133=0{-\frac{8}{3}x^2 + \frac{5}{3}x - \frac{13}{3} = 0}

Q: How do I multiply both sides by -3?

A: To multiply both sides by -3, we need to multiply each term by -3.

Q: What is the result of multiplying both sides by -3?

A: The result of multiplying both sides by -3 is:

8x2−5x+13=0{8x^2 - 5x + 13 = 0}

Q: How do I solve the quadratic equation?

A: To solve the quadratic equation, we can use the quadratic formula:

x=−b±b2−4ac2a{x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}

Q: What are the values of a, b, and c?

A: The values of a, b, and c are:

a = 8 b = -5 c = 13

Q: How do I plug in the values into the quadratic formula?

A: To plug in the values into the quadratic formula, we need to substitute the values of a, b, and c into the formula.

Q: What is the result of plugging in the values into the quadratic formula?

A: The result of plugging in the values into the quadratic formula is:

x=−(−5)±(−5)2−4(8)(13)2(8){x = \frac{-(-5) \pm \sqrt{(-5)^2 - 4(8)(13)}}{2(8)}}

Q: How do I simplify the expression?

A: To simplify the expression, we need to combine like terms.

Q: What is the result of simplifying the expression?

A: The result of simplifying the expression is:

x=5±25−41616{x = \frac{5 \pm \sqrt{25 - 416}}{16}}

Q: How do I simplify the expression further?

A: To simplify the expression further, we need to simplify the square root.

Q: What is the result of simplifying the expression further?

A: The result of simplifying the expression further is:

x=5±−39116{x = \frac{5 \pm \sqrt{-391}}{16}}

Q: How do I simplify the square root?

A: To simplify the square root, we need to express the negative value as a complex number.

Q: What is the result of simplifying the square root?

A: The result of simplifying the square root is:

x=5±i39116{x = \frac{5 \pm i\sqrt{391}}{16}}

Q: How do I simplify the complex number?

A: To simplify the complex number, we need to express the complex number in the form a + bi.

Q: What is the result of simplifying the complex number?

A: The result of simplifying the complex number is:

x=516±i39116{x = \frac{5}{16} \pm \frac{i\sqrt{391}}{16}}

The final answer is 516±i39116\boxed{\frac{5}{16} \pm \frac{i\sqrt{391}}{16}}.