To rewrite the equation \( x^2 + 3x = 6x + 10 \) into a standard quadratic form, we first move all the terms to one side of the equation. We can do this by subtracting \( 6x \) and \( 10 \) from both sides:
\[ x^2 + 3x - 6x - 10 = 0 \]
Now, combine like terms:
\[ x^2 - 3x - 10 = 0 \]
Next, we will factor the quadratic equation \( x^2 - 3x - 10 \). We need two numbers that multiply to \(-10\) (the constant term) and add up to \(-3\) (the coefficient of the linear term). The numbers that satisfy this are \( -5 \) and \( 2 \).
Thus, we can factor the equation as follows:
\[ (x - 5)(x + 2) = 0 \]
Now, we can solve for \( x \) by setting each factor equal to zero:
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\( x - 5 = 0 \) \[ x = 5 \]
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\( x + 2 = 0 \) \[ x = -2 \]
Therefore, the solutions to the equation are:
\[ x = 5 \quad \text{and} \quad x = -2 \]