When completing the square, the above formula is used where \(a\), \(b\), and \(c\) are variables from the equation \(ax^2 + bx + c = 0\).
To complete the square, the formula used is:
\[ ax^2 + bx + c = a \left( x + \frac{b}{2a} \right)^2 - \frac{b^2 - 4ac}{4a} \]
Completing the square is a mathematical technique to rewrite a quadratic equation in a specific form. By rearranging the equation, it allows us to easily identify the vertex or turning point of the parabola represented by the equation.
To complete the square, we take a quadratic equation as \(ax^2 + bx + c\) and manipulate it to create a perfect square trinomial. This is done by adding or subtracting a constant term to both sides of the equation, which is calculated using a specific formula involving the coefficient of the middle term.
The resulting quadratic equation in completed square form, \(a(x - h)^2 + k\), clearly reveals the coordinates \((h, k)\) of the vertex of the parabola. The value of \(h\) represents the horizontal shift from the origin, while \(k\) indicates the vertical shift. This information is crucial as it provides insight into the shape, location, and orientation of the parabola.
Completing the square is important because it simplifies graphing quadratic equations. By converting the equation into completed square form, we can easily determine the vertex, a key point on the graph. This lets us quickly sketch the parabola and understand its behavior without plotting numerous points.
Let's assume the following quadratic equation:
\[ 2x^2 + 8x + 5 = 0 \]
Step 1: Divide all terms by the coefficient of \(x^2\) (which is 2 in this case):
\[ x^2 + 4x + \frac{5}{2} = 0 \]
Step 2: Move the constant term to the other side:
\[ x^2 + 4x = -\frac{5}{2} \]
Step 3: Add the square of half the coefficient of \(x\) to both sides:
\[ x^2 + 4x + 4 = -\frac{5}{2} + 4 \]
Step 4: Simplify the right side:
\[ x^2 + 4x + 4 = \frac{3}{2} \]
Step 5: Write the left side as a square:
\[ (x + 2)^2 = \frac{3}{2} \]
So, the completed square form of the quadratic equation is:
\[ (x + 2)^2 = \frac{3}{2} \]
