The equation 24 x 2 + 25 x − 47 a x − 2

To solve for the value of $a$, we analyze the equation:

$24x^2 + 25x - 47a(x - 2) = -8x - 3 - 53a(x - 2),$ which is true for all $x \neq 2a$.

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Step 1: Simplify the equation

Combine like terms and isolate terms involving $x$ on one side:

Expand:

• Left-hand side: $24x^2 + 25x - 47a(x - 2) = 24x^2 + 25x - 47ax + 94a$
• Right-hand side: $-8x - 3 - 53a(x - 2) = -8x - 3 - 53ax + 106a$

The equation becomes: $24x^2 + 25x - 47ax + 94a = -8x - 3 - 53ax + 106a.$

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Step 2: Rearrange terms

Move all terms to one side to compare coefficients: $24x^2 + (25 - 47a + 8)x + (94a + 3 - 106a) = 0.$

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Step 3: Coefficient comparison

Since the equation is true for all $x$, the coefficients of like powers of $x$ must be equal. This gives us:

1. Coefficient of $x^2$: $24 = 0$ (no contradiction; it simplifies the equation).
2. Coefficient of $x$: $25 - 47a + 8 = 0$.
3. Constant term: $94a + 3 - 106a = 0$.

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Step 4: Solve for $a$

From the coefficient of $x$:

$25 + 8 - 47a = 0 \implies 33 = 47a \implies a = \frac{33}{47}.$

From the constant term:

$94a + 3 - 106a = 0 \implies -12a + 3 = 0 \implies 12a = 3 \implies a = \frac{1}{4}.$

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Final Value of $a$:

This value satisfies the given conditions