Simple Examples Involving Functions of the Roots of a Quadratic Equation
SIMPLE EXAMPLES INVOLVING FUNCTIONS OF THE ROOTS OF A QUADRATIC EQUATION
Understanding the Basic Function of Roots
- The roots of a quadratic equation, often denoted as α and β, are the key points that a parabola intersects with the x-axis.
- Any quadratic equation ax² + bx + c = 0 has roots found using the quadratic formula: x = [-b ± sqrt(b² - 4ac)] / 2a.
Established Relationships Involving Roots
- In a quadratic equation, the sum of the roots α + β = -b/a and the product of the roots αβ = c/a.
- These expressions are directly derived from the coefficients a, b, and c of the standard form of a quadratic equation.
- These relationships can be inverted to construct a quadratic equation from known roots.
Example: Given the Roots
- If given both roots of a quadratic, it’s simple to construct the equation by substitifying into the formula: x² - (sum of roots)x + (product of roots) = 0.
- If α and β are roots, the quadratic equation could be written as x² - (α + β)x + αβ = 0.
Example: Given Fewer Details About the Roots
- Sometimes, only the sum or product of the roots might be given, or neither.
- In these instances, use the given information and the relationships above to find as much information as possible.
- Even if the roots cannot be fully determined, this often leads to further insights about the problem at hand.
Solving Problems with Roots
- Algebraic manipulation can often reveal further information about the roots and their relationships.
- For instance, consider the quadratic equation with roots α and β. What if the roots of another equation were 2α and 2β?
- The new quadratic would lack the exact roots of the previous one, but its relationships (sum and product) could be determined in terms of α and β.
- Remember to connect everything back to the coefficients of the standard form quadratic, as that’s where most of the key insights will originate.
- Use a systematic approach when solving: start simple, and add complexity as needed.