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Why does the same method of completing the square fail when the value of "a" does not equal 1 for a quadratic? ax^2 + bx + c
Reiny
answered
the method of " completing the square " works for all values of a except 0.
In that case the ax^2 term would drop out, and you would not have a quadratic.
I don't know what "same method" you are referring to.
Here is the standard method applied to a specific case
e.g let a = 3/4, such as
y = (3/4)x^2 + (5/8)x + 2/7
= (3/4)(x^2 + (5/6)x + .... - ...) + 2/7
= (3/4)(x^2 + (5/6)x + 25/144 - 25/144) + 2/7
= (3/4)( (x + 5/12)^2 - 25/144) + 2/7
= (3/4)(x+5/12)^2 - 25/192 + 2/7
= (3/4)(x + 5/12)^2 + 209/1344
anonymous
answered
Oh, the "same method" is probably referring to this:
4. Write each quadratic in vertex form by completing the square. Then identify the vertex and solve the quadratic to find the x intercepts.
Explain Bot
answered
The method of completing the square is a technique used to solve quadratic equations of the form ax^2 + bx + c = 0, where "a" is the coefficient of the x^2 term, "b" is the coefficient of the x term, and "c" is the constant term.
When "a" is equal to 1 (meaning the coefficient of x^2 is 1), completing the square works because it allows us to rewrite the quadratic equation as a perfect square trinomial, which can be easily factored.
To complete the square, we take the coefficient of the x term (b), divide it by 2, square the result, and add it to both sides of the equation. This creates a perfect square trinomial on the left side of the equation, which can then be factored into a binomial squared.
However, when "a" is not equal to 1, the same method fails because the process of completing the square becomes more complicated. When "a" is not 1, the coefficient of x^2 affects the entire equation, not just the x^2 term. Adding the same value to both sides of the equation (as is done in completing the square) would cause a change in the constant term, leading to an incorrect equation.
To solve a quadratic equation when "a" is not equal to 1, we generally use other techniques such as factoring (if possible), the quadratic formula, or graphing. These methods are specifically designed to handle quadratic equations in general form and do not rely on completing the square.