Sum and product of the roots of a quadratic equation
We learned on the previous page (The Quadratic Formula), in general there are two roots for any quadratic equation `ax^2+ bx + c = 0`. Let's denote those roots `alpha` and `beta`, as follows:
`alpha=(-b+sqrt(b^2-4ac))/(2a)` and
`beta=(-b-sqrt(b^2-4ac))/(2a)`
Sum of the roots α and β
We can add `alpha` and `beta` as follows:
`alpha + beta =(-b+sqrt(b^2-4ac))/(2a)+(-b-sqrt(b^2-4ac))/(2a)`
` =(-2b+0)/(2a)`
` =-b/a`
Product of the roots α and β
We can multiply `alpha` and `beta` as follows. First, recall that in general,
`(X+Y)(X-Y) = X^2 - Y^2` and
`(sqrt(X))^2 = X`
We make use of these to obtain:
`alpha xx beta = (-b+sqrt(b^2-4ac))/(2a) xx (-b-sqrt(b^2-4ac))/(2a)`
` =((-b)^2 - (sqrt(b^2-4ac))^2)/(2a)^2`
` =(b^2 - (b^2 - 4ac) )/(4a^2)`
` =(4ac)/(4a^2)`
` =c/a`
Summary
The sum of the roots `alpha` and `beta` of a quadratic equation are:
`alpha + beta = -b/a`
The product of the roots `alpha` and `beta` is given by:
`alpha beta = c/a`
It's also important to realize that if `alpha` and `beta` are roots, then:
`(x-alpha)(x-beta)=0`
We can expand the left side of the above equation to give us the following form for the quadratic formula:
`x^2 - (alpha+beta)x + alpha beta = 0`
Let's use these results to solve a few problems.
Example 1
The quadratic equation `2x^2- 7x - 5 = 0` has roots `alpha` and `beta`. Find:
(a) `alpha + beta`
(b) `alpha beta
(c) `alpha^2 + beta^2`
(d) `1/alpha + 1/beta`
Answer
For the expression `2x^2- 7x - 5`, we have:
`a=2`
`b=-7`
`c=-5`
(a) We learned just now that `alpha + beta = -b/a` so in this example,
`alpha + beta = -((-7))/2 = 3.5`
(b) We know `alpha beta = c/a` so in this example,
`alpha beta = (-5)/2 = -2.5`
(c) For `alpha^2 + beta^2`, we need to recall that
`(alpha + beta)^2 = alpha^2 + 2alpha beta + beta^2.`
Solving this for `alpha^2 + beta^2` gives us:
`alpha^2 + beta^2 = (alpha + beta)^2 - 2alpha beta`.
We've already found the sum and product of `alpha` and `beta`, so we can substitute as follows:
`alpha^2 + beta^2 = (3.5)^2 - 2xx(-2.5) = 17.25`.
(d) We add our fractions `1/alpha + 1/beta` as follows:
`1/alpha + 1/beta = (beta + alpha)/(alpha beta) = (alpha + beta)/(alpha beta)`
We know the sum (top) and product (bottom), so we can simply write:
`1/alpha + 1/beta = (alpha + beta)/(alpha beta) = 3.5/(-2.5) = -1.4`
Example 2
Find the quadratic equation with roots α and β given α − β = 2 and α2 − β2 = 3.
Answer
We'll set up a system of two equations in two unknowns to find `alpha` and `beta`.
Remembering the difference of squares formula, we have
α2 − β2 = (α + β)(α − β)
From the question we know α2 − β2 = 3, so this gives us:
3 = (α + β)(α − β)
The question says α − β = 2, which we can substitute into the right hand side, giving:
3 = 2(α + β)
This gives:
`(alpha + beta) = 3/2`
Using α − β = 2 again, we add it to the above line, giving:
`2 alpha = 3/2 + 2 = 7/2`
So `alpha = 7/4`
Since `(alpha + beta) = 3/2` then `beta = 3/2 - alpha`, giving us `beta = -1/4`.
We substitute these values into the expression `x^2 - (alpha+beta)x + alpha beta = 0` giving:
`x^2 - (3/2)x + (7/4)(-1/4) = 0`
`x^2 -3/2 x -7/16=0`
So the required quadratic equation is:
`x^2 -3/2 x -7/16 = 0`
We multiply throughout by `16` to tidy it up:
`16x^2 - 24x - 7 = 0`
Let's now go on to learn how the graph of a quadratic function is a parabola: 4. The Graph of the Quadratic Function
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