Roots of a Quadratic Equation

UNDERSTANDING THE ROOTS OF A QUADRATIC EQUATION

• The roots of a quadratic equation are the x-values where the graph of the function intersects with the x-axis.
• The quadratic function f(x) = ax² + bx + c has roots found by using the quadratic formula: x = [-b ± sqrt(b² - 4ac)] / 2a.
• When calculating roots, always start by simplifying the quadratic equation as much as possible, bringing it to its standard form ax² + bx + c = 0.

THE DISCRIMINANT AND ROOTS

• The Discriminant, calculated as b² - 4ac, determines the number and type of roots in a quadratic equation.
• If b² - 4ac > 0, there are two distinct real roots.
• If b² - 4ac = 0, there is exactly one real root, also known as a repeated root.
• If b² - 4ac < 0, there are two complex roots. This means the graph of the quadratic function does not intersect the x-axis.

EFFECTS OF ROOTS ON PARABOLA SHAPE

• If a quadratic function has two distinct real roots, the parabola crosses the x-axis at two points.
• If it has one real root, the vertex of the parabola touches the x-axis.
• If it has two complex roots, the parabola does not intersect or touch the x-axis.
• Hence, the characteristics and position of the roots can provide crucial information about the shape and position of the parabola.

SUM AND PRODUCT OF ROOTS

• For any quadratic equation f(x) = ax² + bx + c = 0, if α and β are the roots of the equation, then sum of the roots = -b/a.
• Similarly, for the same quadratic equation, the product of the roots = c/a.
• These formulas can be used to form a quadratic equation when the roots are known, or to find the roots when the sum and product are known.

SOLVING QUADRATIC EQUATIONS

• Quadratic equations can be solved by factoring, completing the square, or using the quadratic formula.
• Choosing between these methods often depends on the nature of the coefficients and the term b² - 4ac.
• Working out problems using all three methods can often help to deepen understanding of quadratic equations and their roots.