How do you verify if a calculated root of a quadratic equation is correct?

Conditions

• You have a candidate solution for $x$.
• The original quadratic equation is available.

Reasoning, step by step

1. Take the calculated root value.
2. Substitute it into the original equation wherever $x$ appears.
3. Calculate the exponent (square) of the substituted value.
4. Perform the multiplications.
5. Combine the terms using addition and subtraction.
6. Check if the final result equals zero.

Example

To verify $x = -2$ for the equation $2x^2 + 3x - 2 = 0$, substitute $-2$ for $x$: $2(-2)^2 + 3(-2) - 2$. This simplifies to $2(4) - 6 - 2 = 8 - 8 = 0$. Since $0 = 0$, the root is valid.

Common misconceptions

• Forgetting to apply the exponent to the negative sign correctly (e.g., calculating $-2^2$ as $-4$ instead of $4$).
• Assuming that only one root needs to be verified to prove the entire solution set is correct.

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