To solve the quadratic equation \( x^2 - 12x + 51 = 2x + 3 \), we first move all the terms to one side of the equation to set it to zero:
\[ x^2 - 12x + 51 - 2x - 3 = 0 \]
Now, combine like terms:
\[ x^2 - 14x + 48 = 0 \]
Next, we will factor the quadratic equation. We are looking for two numbers that multiply to \( 48 \) (the constant term) and add up to \( -14 \) (the coefficient of \( x \)).
The numbers \( -6 \) and \( -8 \) fit these criteria:
\[ -6 + (-8) = -14 \ -6 \times -8 = 48 \]
Now we can factor the quadratic:
\[ (x - 6)(x - 8) = 0 \]
To find the solutions for \( x \), we set each factor equal to zero:
- \( x - 6 = 0 \) leads to \( x = 6 \)
- \( x - 8 = 0 \) leads to \( x = 8 \)
Thus, the solutions to the equation are:
\[ x = 6 \quad \text{and} \quad x = 8 \]
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