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Sagot :
To complete the proof of the quadratic formula and fill in the missing reason in the last step of your table, follow these steps carefully:
### Proof Steps:
1. Given:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
2. Subtract [tex]\( c \)[/tex] from both sides:
[tex]\[ ax^2 + bx = -c \][/tex]
Reason: Subtract [tex]\( c \)[/tex] from both sides of the equation.
3. Divide both sides by [tex]\( a \)[/tex]:
[tex]\[ x^2 + \frac{b}{a} x = -\frac{c}{a} \][/tex]
Reason: Divide both sides of the equation by [tex]\( a \)[/tex].
4. Complete the square and add [tex]\( \left(\frac{b}{2a}\right)^2 \)[/tex] to both sides:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 \][/tex]
Reason: Complete the square and add [tex]\( \left(\frac{b}{2a}\right)^2 \)[/tex] to both sides of the equation.
5. Square [tex]\( \left(\frac{b}{2a}\right) \)[/tex] on the right side of the equation:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} \][/tex]
Reason: Square [tex]\( \left(\frac{b}{2a}\right) \)[/tex] on the right side of the equation.
6. Find a common denominator on the right side of the equation:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} \][/tex]
Reason: Find a common denominator on the right side of the equation.
7. Add the terms on the right side of the equation:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \][/tex]
Reason: Add the terms on the right side of the equation.
### Complete Table:
[tex]\[ \begin{array}{|l|l|} \hline \text{Statements} & \text{Reasons} \\ \hline ax^2 + bx + c = 0 & \text{Given} \\ \hline ax^2 + bx = -c & \text{Subtract } c \text{ from both sides of the equation} \\ \hline x^2 + \frac{b}{a} x = -\frac{c}{a} & \text{Divide both sides of the equation by } a \\ \hline x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 & \text{Complete the square and add } \left(\frac{b}{2a}\right)^2 \text{ to both sides} \\ \hline x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} & \text{Square } \left(\frac{b}{2a}\right) \text{ on the right side of the equation} \\ \hline x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} & \text{Find a common denominator on the right side of the equation} \\ \hline x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} & \text{Add the terms on the right side of the equation} \\ \hline \end{array} \][/tex]
### Proof Steps:
1. Given:
[tex]\[ ax^2 + bx + c = 0 \][/tex]
2. Subtract [tex]\( c \)[/tex] from both sides:
[tex]\[ ax^2 + bx = -c \][/tex]
Reason: Subtract [tex]\( c \)[/tex] from both sides of the equation.
3. Divide both sides by [tex]\( a \)[/tex]:
[tex]\[ x^2 + \frac{b}{a} x = -\frac{c}{a} \][/tex]
Reason: Divide both sides of the equation by [tex]\( a \)[/tex].
4. Complete the square and add [tex]\( \left(\frac{b}{2a}\right)^2 \)[/tex] to both sides:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 \][/tex]
Reason: Complete the square and add [tex]\( \left(\frac{b}{2a}\right)^2 \)[/tex] to both sides of the equation.
5. Square [tex]\( \left(\frac{b}{2a}\right) \)[/tex] on the right side of the equation:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} \][/tex]
Reason: Square [tex]\( \left(\frac{b}{2a}\right) \)[/tex] on the right side of the equation.
6. Find a common denominator on the right side of the equation:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} \][/tex]
Reason: Find a common denominator on the right side of the equation.
7. Add the terms on the right side of the equation:
[tex]\[ x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} \][/tex]
Reason: Add the terms on the right side of the equation.
### Complete Table:
[tex]\[ \begin{array}{|l|l|} \hline \text{Statements} & \text{Reasons} \\ \hline ax^2 + bx + c = 0 & \text{Given} \\ \hline ax^2 + bx = -c & \text{Subtract } c \text{ from both sides of the equation} \\ \hline x^2 + \frac{b}{a} x = -\frac{c}{a} & \text{Divide both sides of the equation by } a \\ \hline x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \left(\frac{b}{2a}\right)^2 & \text{Complete the square and add } \left(\frac{b}{2a}\right)^2 \text{ to both sides} \\ \hline x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{c}{a} + \frac{b^2}{4a^2} & \text{Square } \left(\frac{b}{2a}\right) \text{ on the right side of the equation} \\ \hline x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = -\frac{4ac}{4a^2} + \frac{b^2}{4a^2} & \text{Find a common denominator on the right side of the equation} \\ \hline x^2 + \frac{b}{a} x + \left(\frac{b}{2a}\right)^2 = \frac{b^2 - 4ac}{4a^2} & \text{Add the terms on the right side of the equation} \\ \hline \end{array} \][/tex]
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