A quadratic equation can be written in many different forms. The intercept form of a quadratic equation looks like a(x - p)(x - q) = 0. You can see that a quadratic equation in the intercept form is synonymous with its factorized form. Thus, in order to convert a quadratic equation to the intercept form, you have to factorize it.
Quadratic equations can be factorized by the method of splitting their middle terms or the box method, both discussed extensively in their particular posts. Please visit the individual posts by clicking on their links above to get further insight into these methods.
Let us take an example of a quadratic equation and convert it into its intercept form. We'll start with a quadratic equation written in its standard form, and by applying a few steps as outlined below, convert it into its intercept form:
Quadratic equation in standard form: x2 + 2x - 3 = 0
Steps for converting the above equation to its intercept form:
Step 1: Split the middle term of the quadratic expression
In the above method, we have simply factorized a quadratic expression and written it as a product of two linear expressions. The last equation obtained, (x - 1)(x + 3) = 0, is a quadratic equation in its intercept form. The original equation taken, x2 + 2x - 3 = 0, is a quadratic equation that could be factored, but there are quadratic equations, like 2x2 + 4x - 5 = 0, in which the quadratic expression is a prime one and it cannot be factored. Hence prime quadratic equations can not be converted to their intercept forms as they do not get factored.
Quadratic equations can be factorized by the method of splitting their middle terms or the box method, both discussed extensively in their particular posts. Please visit the individual posts by clicking on their links above to get further insight into these methods.
Let us take an example of a quadratic equation and convert it into its intercept form. We'll start with a quadratic equation written in its standard form, and by applying a few steps as outlined below, convert it into its intercept form:
Quadratic equation in standard form: x2 + 2x - 3 = 0
Steps for converting the above equation to its intercept form:
Step 1: Split the middle term of the quadratic expression
x2 + 3x - x - 3 = 0Step 2: Take common factors from each group of two factors
x(x + 3) - 1(x + 3) = 0Step 3: Take (x + 3) as the common factor
(x - 1)(x + 3) = 0The above quadratic equation is in its intercept form. Comparing (x - 1)(x + 3) = 0 with a(x - p)(x - q) = 0, we obtain a = 1, p = 1 and q = -3.
In the above method, we have simply factorized a quadratic expression and written it as a product of two linear expressions. The last equation obtained, (x - 1)(x + 3) = 0, is a quadratic equation in its intercept form. The original equation taken, x2 + 2x - 3 = 0, is a quadratic equation that could be factored, but there are quadratic equations, like 2x2 + 4x - 5 = 0, in which the quadratic expression is a prime one and it cannot be factored. Hence prime quadratic equations can not be converted to their intercept forms as they do not get factored.
Thank you sooo much, our teacher gave us problems like these for hw over winterbreak w/o teaching us how to do it and we have a test the day we get back!
ReplyDeleteWe can convert a quadratic function, in standard form y = ax^2 + bx + c, into the intercept form.
ReplyDeletey = a*(x - x1)(x - x2)
y = a*(x^2 + bx/a + c/a) (1)
Recall the development of the quadratic formula:
x^2 + bx/a + c/a + (b^2/4a^2 - b^2/4a^2) = 0
(x + b/2a)^2 - (b^2 - 4ac)/4a^2 = (x + b/2a)^2 - d^2/4a^2. (2)
Call d^2 = b^2 - 4ac. Replace this expression (2) into (1), we get the quadratic function written in intercept form:
y = a*(x + b/2a + d/2a)(x + b/2a - d/2a) (3) with d^2 = b^2 - 4ac.
From this form, we deduct the Quadratic Formula in Intercept Form:
x = -b/2a + (or -) d/2a. (4)
In this formula:
- the quantity (-b/2a) represents the x-coordinate of the parabola axis.
- The 2 quantities (-d/2a) and (d/2a) represent the 2 equal distances from the axis to the two x-intercepts.
- If d = 0, there is double root at x = -b/2a.
- If d^2 > 0, There are 2 real roots (2 x-intercepts)
- If d^2 < 0, there are no intercepts. There are 2 complex roots.