To complete the square for the quadratic equation, we will simplify the steps as follows:
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Factor out the coefficient from the first two terms:
If the quadratic equation is in the form \( ax^2 + bx + c \), we factor out \( a \) from the first two terms, resulting in \( a(x^2 + \frac{b}{a}x) + c \). -
Complete the square inside the parentheses:
- Take the coefficient of \( x \) (which is \( \frac{b}{a} \)), halve it, yielding \( \frac{b}{2a} \), and then square it, resulting in \( \left( \frac{b}{2a} \right)^2 \).
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Add and subtract this square inside the parentheses:
- The expression becomes:
\( a\left( x^2 + \frac{b}{a}x + \left( \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 \right) + c \).
- The expression becomes:
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Distribute the negative sign:
- This results in:
\( a\left( \left( x + \frac{b}{2a} \right)^2 - \left( \frac{b}{2a} \right)^2 \right) + c \). - Simplifying, we have:
\( a\left( x + \frac{b}{2a} \right)^2 - a\left( \frac{b}{2a} \right)^2 + c \).
- This results in:
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Combine constants:
- This leads to the completed square form:
\( a\left( x + \frac{b}{2a} \right)^2 + \left(c - \frac{b^2}{4a}\right) \).
- This leads to the completed square form:
From this completed square form, we can identify the following:
- The vertex of the parabola is at \( \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right) \).
- The axis of symmetry is the vertical line \( x = -\frac{b}{2a} \).
In summary, from our completed square form, we have:
- The vertex is \( \left(-\frac{b}{2a}, c - \frac{b^2}{4a}\right) \).
- The axis of symmetry is \( x = -\frac{b}{2a} \).