To identify what George might have done wrong without seeing the table, we can analyze the given quadratic equation step by step.
First, let's simplify the equation. The original equation is:
\[ x^2 + 4x - 5 = 7 \]
To solve for \(x\), we need to move all terms to one side of the equation:
- Subtract 7 from both sides: \[ x^2 + 4x - 5 - 7 = 0 \] \[ x^2 + 4x - 12 = 0 \]
Now, we have the standard form of the quadratic equation \( ax^2 + bx + c = 0 \), where:
- \( a = 1 \)
- \( b = 4 \)
- \( c = -12 \)
Next, we can solve this equation using the quadratic formula, which is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Calculating the discriminant:
- \( b^2 - 4ac = 4^2 - 4(1)(-12) = 16 + 48 = 64 \)
Now we can find the values of \(x\): \[ x = \frac{-4 \pm \sqrt{64}}{2(1)} = \frac{-4 \pm 8}{2} \]
This gives us two solutions:
- \( x = \frac{-4 + 8}{2} = \frac{4}{2} = 2 \)
- \( x = \frac{-4 - 8}{2} = \frac{-12}{2} = -6 \)
Thus, the solutions to the equation are \( x = 2 \) and \( x = -6 \).
Possible Errors
Now, if George provided a table of his work, he might have made several common errors, such as:
- Miscalculating the equation after moving terms.
- Incorrectly applying the quadratic formula.
- Making arithmetic mistakes in calculating \( b^2 - 4ac \).
- Improperly simplifying the solutions.
To give more specific feedback on what George did wrong, I would need to see his table and work. If you can provide that information, I can diagnose the error more accurately!
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