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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`
- 5. Equations in Quadratic Form
5. Equations in Quadratic Form In this section, ...
- Interactive Quadratic Function Graph
Interactive Quadratic Function Graph In the previous ...
- 1. Solving Quadratic Equations by Factoring
Summary In general, a quadratic equation: must contain ...
- Parabola
In general, the graph of a quadratic equation `y = ...
- Completing the Square
follow these steps: (i) If a does not equal `1`, divide ...
- Quadratic Formula
3. The Quadratic Formula At the end of the last section ...
- 5. Equations in Quadratic Form
2017年9月19日 · Considering the following equation $$px^2 - qx - r = 0$$ whose roots are $\alpha$ and $β$. What would $α - β$ be in terms of $p$, $q$, and $r$? I understand that I have to use the sum and product of roots, and have found both in terms of $p$, $q$, and $r$:
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Identity 1: α 2 + β 2. α2 + β2 = (α + β)2 − 2αβ. Deriving the formula: Since (a + b)2 = a2 + 2ab + b2, (α + β)2 = α2 + 2αβ + β2 (α + β)2 − 2αβ = α2 + β2 ∴ α2 + β2 = (α + β)2 − 2αβ. Identity 2: (α - β) 2. (α − β)2 = (α + β)2 − 4αβ. Deriving the formula:
Vieta's formula relates the coefficients of polynomials to the sums and products of their roots, as well as the products of the roots taken in groups. For example, if there is a quadratic polynomial \ (f (x) = x^2+2x -15 \), it will have roots of \ (x=-5\) and \ (x=3\), because \ (f (x) = x^2+2x-15= (x-3) (x+5)\).
The roots of the quadratic function y = 1 2 x2 − 3x + 5 2 are the places where the graph intersects the x -axis, the values x = 1 and x = 5. They can be found via the quadratic formula. In elementary algebra, the quadratic formula is a closed-form expression describing the solutions of a quadratic equation.
The roots of a quadratic equation are usually represented to by the symbols alpha (α), and beta (β). Here we shall learn more about how to find the nature of roots of a quadratic equation without actually finding the roots of the equation.
