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When deriving the quadratic formula, particularly when working on completing the square, we aim to transform the equation into a perfect square trinomial on one side. The given steps in the derivation can be broken down as follows:
1. Start with the original quadratic equation and arrange it such that the constant term is separated:
[tex]\[ -c = ax^2 + bx \][/tex]
Here, we use the subtraction property of equality by moving \(c\) to the left-hand side.
2. Factor out the coefficient \(a\) from the terms involving \(x\):
[tex]\[ -c = a \left( x^2 + \frac{b}{a} x \right) \][/tex]
3. Determine the value needed to complete the square. Half of the coefficient of \(x\) (which is \(\frac{b}{a}\)) is \(\frac{b}{2a}\). Squaring this value gives:
[tex]\[ \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} \][/tex]
4. To complete the square on the right side of the equation, we add and subtract \(\frac{b^2}{4a}\) inside the equation:
[tex]\[ -c + \frac{b^2}{4a} - a \left( x^2 + \frac{b}{a} x + \frac{b^2}{4a^2} \right) \][/tex]
Here, \(\frac{b^2}{4a}\) is added to balance the trinomial on the right side. But since adding this term changes the balance of the equation, we need to subtract the same \(\frac{b^2}{4a}\) term from the left-hand side. This subtraction keeps the equation balanced.
5. The key point in the final step is to maintain the equality of the equation. By adding \(\frac{b^2}{4a^2}\) to the right side, we must subtract an equivalent term from the left side to preserve equality. This balancing ensures the equation remains correct through the completion of the square process.
Therefore, the most accurate explanation is:
- The term \(\frac{b^2}{4a^2}\) is added to the right side of the equation, so it needs to be subtracted from the left side of the equation to balance the sides of the equation.
Thus, the correct answer is:
- The term [tex]\(\frac{b^2}{4 a^2}\)[/tex] is added to the right side of the equation, so it needs to be subtracted from the left side of the equation to balance the sides of the equation.
1. Start with the original quadratic equation and arrange it such that the constant term is separated:
[tex]\[ -c = ax^2 + bx \][/tex]
Here, we use the subtraction property of equality by moving \(c\) to the left-hand side.
2. Factor out the coefficient \(a\) from the terms involving \(x\):
[tex]\[ -c = a \left( x^2 + \frac{b}{a} x \right) \][/tex]
3. Determine the value needed to complete the square. Half of the coefficient of \(x\) (which is \(\frac{b}{a}\)) is \(\frac{b}{2a}\). Squaring this value gives:
[tex]\[ \left(\frac{b}{2a}\right)^2 = \frac{b^2}{4a^2} \][/tex]
4. To complete the square on the right side of the equation, we add and subtract \(\frac{b^2}{4a}\) inside the equation:
[tex]\[ -c + \frac{b^2}{4a} - a \left( x^2 + \frac{b}{a} x + \frac{b^2}{4a^2} \right) \][/tex]
Here, \(\frac{b^2}{4a}\) is added to balance the trinomial on the right side. But since adding this term changes the balance of the equation, we need to subtract the same \(\frac{b^2}{4a}\) term from the left-hand side. This subtraction keeps the equation balanced.
5. The key point in the final step is to maintain the equality of the equation. By adding \(\frac{b^2}{4a^2}\) to the right side, we must subtract an equivalent term from the left side to preserve equality. This balancing ensures the equation remains correct through the completion of the square process.
Therefore, the most accurate explanation is:
- The term \(\frac{b^2}{4a^2}\) is added to the right side of the equation, so it needs to be subtracted from the left side of the equation to balance the sides of the equation.
Thus, the correct answer is:
- The term [tex]\(\frac{b^2}{4 a^2}\)[/tex] is added to the right side of the equation, so it needs to be subtracted from the left side of the equation to balance the sides of the equation.
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