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Rewrite the following equation in a quadratic equation and solve by factoring: d^2 + 3x = 6x + 10. Which of the following contains the correct values for x?

Which answer is it?:
1. 2, 5
2. -2, -5
3. 2, -5
4. -2, 5

1 answer

To rewrite the equation \( d^2 + 3x = 6x + 10 \) in the form of a quadratic equation, we first rearrange the terms:

  1. Move all terms to one side of the equation: \[ d^2 + 3x - 6x - 10 = 0 \] This simplifies to: \[ d^2 - 3x - 10 = 0 \]

  2. Rearranging gives us the standard form: \[ d^2 - 3x - 10 = 0 \]

Next, we can factor this equation. We look for two numbers that multiply to \(-10\) (the constant term) and add to \(-3\) (the coefficient of \(x\)). The numbers that fulfill these requirements are \(-5\) and \(2\).

Factoring gives us: \[ (d - 5)(d + 2) = 0 \]

  1. Setting each factor to zero gives us the solutions for \(d\): \[ d - 5 = 0 \quad \Rightarrow \quad d = 5 \] \[ d + 2 = 0 \quad \Rightarrow \quad d = -2 \]

Thus, the values for \(d\) (which corresponds to \(x\) in your options) are \(5\) and \(-2\).

Finally, looking at the provided options, the correct values for \(x\) are \( -2 \) and \( 5 \), which corresponds to:

Option 4: -2, 5.