- Solve The Equation: Solve The Equation: 5(5x 2) + 3x = 28(x - 1)x = 121I
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Introduction to the Equation
Understanding how to approach and solve algebraic equations is fundamental in mathematics. The given equation, 5(5x 2) + 3x = 28(x - 1)x = 121I, appears complex at first glance due to its structure and multiple expressions. To effectively solve it, we need to interpret each part carefully, identify the variables and constants, and simplify step-by-step. This article aims to break down the problem systematically, explore various possible interpretations, and guide you through the process of solving the equation.
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Deciphering the Equation
Analyzing the Given Expression
The original statement is:
> 5(5x 2) + 3x = 28(x - 1)x = 121I
At first glance, it seems to contain some ambiguities, possibly typographical errors or formatting issues. The expression includes:
- A term: 5(5x 2)
- An additional term: 3x
- An equality: = 28(x - 1)x
- A final equality: = 121I
Given the presentation, it seems plausible that the intended equation is:
5(5x + 2) + 3x = 28(x - 1)x = 121I
or perhaps:
5(5x + 2) + 3x = 28(x - 1)x = 121I
However, the presence of "121I" is ambiguous. It could represent:
- 121 multiplied by I (the imaginary unit), or
- A notation error.
In typical algebraic contexts, "I" often denotes the imaginary unit, but in some cases, it might be a variable or constant.
Given the structure, the most logical interpretation is that the equation involves:
- A left-hand expression involving 5(5x + 2) + 3x
- An expression: 28(x - 1)x
- An equality to a value, possibly 121I (or 121 times some variable or constant)
For clarity, let's assume the equation is:
5(5x + 2) + 3x = 28(x - 1)x = 121I
which suggests that:
- The first expression equals the second, and the second equals 121I
Thus, the equation implies:
5(5x + 2) + 3x = 28(x - 1)x = 121I
or equivalently,
5(5x + 2) + 3x = 28(x - 1)x
and
28(x - 1)x = 121I
Given this, our goal is to:
- Find the value(s) of x satisfying the first equality
- Understand the role of 121I, which could involve complex numbers if I is imaginary
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Step 1: Simplify the First Expression
Let's start by simplifying the first part of the equation:
5(5x + 2) + 3x
Applying distributive property:
5 5x + 5 2 + 3x = 25x + 10 + 3x
Combine like terms:
(25x + 3x) + 10 = 28x + 10
So, the first expression simplifies to:
28x + 10
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Step 2: Simplify the Second Expression
Next, consider the second expression:
28(x - 1)x
This is a product of two factors: 28, and (x - 1)x.
First, expand (x - 1)x:
(x - 1) x = x^2 - x
Multiply by 28:
28(x^2 - x) = 28x^2 - 28x
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Step 3: Establish the Main Equation
From the simplifications, the initial set of equalities reduces to:
28x + 10 = 28x^2 - 28x
Since the original statement indicates both expressions are equal (before the "= 121I"), we set:
28x + 10 = 28x^2 - 28x
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Step 4: Formulate a Quadratic Equation
Bring all terms to one side:
28x^2 - 28x - 28x - 10 = 0
Simplify:
28x^2 - 56x - 10 = 0
Divide the entire equation by 2 to simplify coefficients:
14x^2 - 28x - 5 = 0
This quadratic equation is:
14x^2 - 28x - 5 = 0
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Step 5: Solve the Quadratic Equation
Use the quadratic formula:
x = [-b ± sqrt(b^2 - 4ac)] / 2a
where:
- a = 14
- b = -28
- c = -5
Calculate the discriminant:
D = b^2 - 4ac = (-28)^2 - 4 14 (-5) = 784 + 280 = 1064
Since the discriminant is positive, there are two real solutions.
Calculate sqrt(D):
sqrt(1064) ≈ 32.6
Now, apply the quadratic formula:
x = [28 ± 32.6] / (2 14) = [28 ± 32.6] / 28
Calculate both solutions:
- x₁ = (28 + 32.6) / 28 ≈ 60.6 / 28 ≈ 2.164
- x₂ = (28 - 32.6) / 28 ≈ -4.6 / 28 ≈ -0.164
Thus, the solutions to the quadratic are approximately:
- x ≈ 2.164
- x ≈ -0.164
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Step 6: Considering the Implication of 121I
Recall that the original equation included "= 121I." If I is intended as the imaginary unit, then the equation involves complex numbers.
The equation:
28(x - 1)x = 121I
From earlier, we have:
28x^2 - 28x = 121I
Divide both sides by 28:
x^2 - x = (121/28) I
This suggests that:
x^2 - x is a purely imaginary number.
Since the left side is a quadratic expression in x with real coefficients, for the right side to be purely imaginary, x must be complex.
Let’s let:
x = p + iq, where p and q are real numbers.
Calculate:
x^2 - x = (p + iq)^2 - (p + iq) = (p^2 - q^2 + 2ipq) - p - iq
Simplify:
= (p^2 - q^2 - p) + i(2pq - q)
This must equal:
(121/28) I
which is purely imaginary. Therefore, the real part must be zero:
p^2 - q^2 - p = 0
and the imaginary part:
2pq - q = (121/28)
Factor the imaginary part:
q(2p - 1) = 121/28
Now, from the real part:
p^2 - p = q^2
Express p in terms of q:
From the imaginary part:
q = (121/28) / (2p - 1)
Substitute into the real part:
p^2 - p = [(121/28)^2] / (2p - 1)^2
This results in a complex equation in p, which can be solved numerically or graphically. Alternatively, since the problem becomes quite involved, and the primary goal is solving for x, the approximate solutions obtained earlier (x ≈ 2.164 and x ≈ -0.164) are relevant only if the complex component is not considered.
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Summary of Solutions
- If the equation is interpreted purely in real numbers, the solutions are approximately:
- x ≈ 2.164
- x ≈ -0.164
- If the imaginary component is involved, the solutions for x are complex and require solving the system:
- p^2 - p = q^2
- q(2p - 1) = 121/28
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Conclusion
The process of solving the given equation involves several steps:
- Interpreting the notation and structure carefully.
- Simplifying algebraic expressions to reduce the problem to a quadratic.
- Applying the quadratic formula for real solutions.
- Considering complex solutions if the equation involves imaginary units.
By methodically breaking down the problem, we can arrive at approximate real solutions and understand the nature of possible complex solutions. This approach
