Enter the variables a, b, and c into the calculator below to calculate the two solutions to completing the square.
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Completing The Square Formula
When completing the square, the above formula is used where a, b, and c are variables from the equation
ax^2+bx+c=0
Completing the Square Definition
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.
How to calculate completing the square?
How to complete the square
- First, determine the variables
Calculate or gather the variables a,b, and c from an equation of the form ax^2+bx+C.
- Enter the variables into the formula or calculator above.
The solution should be two separate answers, typically the x-intercepts.
FAQ
What is the purpose of completing the square in quadratic equations?
Completing the square is used to transform a quadratic equation into a form that makes it easier to understand the properties of the parabola it represents, such as its vertex and direction of opening. This method simplifies the process of graphing the equation.
Can every quadratic equation be solved by completing the square?
Yes, every quadratic equation can be solved by completing the square. It is a universal method that works for all quadratic equations, although the ease of computation may vary depending on the coefficients of the equation.
What are the advantages of completing the square over using the quadratic formula?
Completing the square provides a more visual insight into the properties of a quadratic equation, such as the location of its vertex. It can be particularly useful in situations where understanding the geometry of the equation is important. While the quadratic formula directly gives the roots, completing the square helps with understanding the equation’s graph.
Is completing the square applicable in real-world problems?
Yes, completing the square is applicable in various real-world problems, especially in physics and engineering, where understanding the motion of objects or optimizing certain parameters is necessary. It’s also used in calculus for integrating certain types of functions.
