Question

Relation between root and coefficient​

Answer 1

The sum of the roots of a quadratic equation is equal to the negation of the coefficient of the second term, divided by the leading coefficient. The product of the roots of a quadratic equation is equal to the constant term (the third term), ... The roots will be represented as r1 and r2.

Answer 2

Let us take the quadratic equation of the general form ax^2 + bx + c = 0 where a (≠ 0) is the coefficient of x^2, b the coefficient of x and c, the constant term.

Let α and β be the roots of the equation ax^2 + bx + c = 0

Now we are going to find the relations of α and β with a, b and c.

Now ax^2 + bx + c = 0

Multiplication both sides by 4a (a ≠ 0) we get

4a^2x^2 + 4abx + 4ac = 0

(2ax)^2 + 2 * 2ax * b + b^2 – b^2 + 4ac = 0

(2ax + b)^2 = b^2 - 4ac

2ax + b = ±√b^2−4ac

x = −b±√b2−4ac/2a

Therefore, the roots of (i) are −b±√b2−4ac/2a

Let α = −b+√b2−4ac/2a and β = −b-√b2−4ac/2a

Therefore,

α + β = −b+√b2−4ac/2a + −b-√b2−4ac/2a

α + β = −2b2a

α + β = -ba

α + β = -coefficient of X /coefficientof x^2

Again, αβ =−b+√b2−4ac/2a× −b-√b2−4ac/2a

αβ = (−b)^2−√(b^2−4ac)^2/ 4a^2

αβ = b^2−(b^2−4ac)/4a^2

αβ = 4ac/4a^2

αβ = ca

αβ = constant term / coefficient of x^2

Therefore, α + β = -coefficient of x / coefficient of x^2 and αβ = constant term / coefficient of x^2 represent the required relations between roots (i.e., α and β) and coefficients (i.e., a, b and c) of equation ax^2 + bx + c = 0.

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