Relationships between the Roots and Coefficients of a Quadratic Equation

Introduction to Roots of Quadratic Equation

A quadratic equation is any equation that can be rewritten in the form ax^2 + bx + c = 0 where a, b, and c are real coefficients, and a ≠ 0.
A quadratic equation has two solutions or roots which may be real, complex, or identical.

The roots of a quadratic equation can directly be related to its coefficients.
If α and β are the roots of the quadratic equation ax^2 + bx + c = 0, then the sum of the roots (α + β) is equal to the negative ratio of the linear coefficient (b) to the leading coefficient (a). Express this as α + β = -b/a.
Similarly, the product of the roots (α * β) is equal to the constant term (c) divided by the leading coefficient (a). Represent this as α * β = c/a.

A quadratic equation will have equal roots if the discriminant (b^2 - 4ac) is equal to zero. The discriminant is part of the quadratic formula x = [-b ± sqrt(b² - 4ac)] / 2a which is used to find the roots of the quadratic equation.
When roots are equal, the quadratic equation can be expressed as a perfect square trinomial, which is in the form (x - α)^2 = 0.

If the discriminant (b^2 - 4ac) is less than zero, the quadratic equation will have complex roots.
Complex roots always come in conjugate pairs. If α = p + iq is a root of the quadratic equation with real coefficients, then its conjugate α’ = p - iq will also be a root.

The coefficients of a quadratic equation provide a wealth of information about its roots.
The sum and product of the roots can be found using the coefficients of the equation.
Evaluating the discriminant allows insights about the nature of the roots.
With these tools, you can successfully decipher and work with a variety of quadratic equations in further mathematics.