Relationship Between Zeroes And Coefficients (Quadratic)

The student can verify the algebraic relationship between the coefficients of a quadratic equation and the sum/product of its calculated zeroes.

8 lessons

0116ed33...

concept

Algebra Meets Coefficients

Introduce the algebraic relationship between zeroes and coefficients.

How do the roots (zeroes) of a quadratic equation relate to its shape? It turns out the coefficients $a$ , $b$ , and $c$ in the polynomial $ax^2 + bx + c$ hold the secret to the sum and product of those zeroes.

1 / 5

34ab62ad...

b90c37f9...

html

Quadratic Zero Relationships

The central formula linking roots to coefficients.

Quadratic Polynomial

p (x) = a x^{2} + b x + c (a \neq = 0)

Let the zeroes of the polynomial be $α$ and $β$ .

Sum of Zeroes

α + β = \frac{- b}{a}

Negative coefficient of x ÷ coefficient of x²

Product of Zeroes

α β = \frac{c}{a}

Constant term ÷ coefficient of x²

fe59f56d...

worked example

Verifying Relationships

Find zeroes by splitting the middle term and verify relations.

Problem. Find the zeroes of the quadratic polynomial $x^2 + 7x + 10$ , and verify the relationship between the zeroes and the coefficients.

1 / 6

715f3085...

worked example

Missing Middle Term

Find zeroes using a^2 - b^2 identity.

Problem. Find the zeroes of the polynomial $x^2 - 3$ and verify the relationship between the zeroes and the coefficients.

1 / 6

fc126dab...

reading

Verify It Yourself

Faded example for verifying relations.

0 of 3 blanks filled

Let's find the zeroes of the quadratic polynomial $4s^2 - 4s + 1$ and verify their relationship with the coefficients. First, we factorize the polynomial by splitting the middle term to get $(2s - 1)(2s - 1) = 0$ . This gives us two identical zeroes, each equal to . Now we verify the sum of the zeroes: $1/2 + 1/2 = 1$ . Using the coefficient formula $-b/a$ , we substitute our values to get $-($ $) / 4 = 1$ , which matches the calculated sum. Finally, we verify the product of the zeroes: $1/2 \times 1/2 = 1/4$ . Using the formula $c/a$ , we substitute the constant term to get / 4 = 1/4, matching the calculated product perfectly.

99de2883...

matching

Match the Sum and Product

Match quadratic polynomials to their sum and product values.

NoteMatch each quadratic polynomial with the correct values for its Sum (-b/a) and Product (c/a).

0 of 4 connected

Terms

Definitions

← previousGeometrical Meaning Of Zeroes next →Forming A Quadratic Polynomial From Zero...

concept

Algebra Meets Coefficients

Introduce the algebraic relationship between zeroes and coefficients.

1 / 5

worked example

Verifying Relationships

Find zeroes by splitting the middle term and verify relations.

Problem. Find the zeroes of the quadratic polynomial $x^2 + 7x + 10$ , and verify the relationship between the zeroes and the coefficients.

1 / 6

worked example

Missing Middle Term

Find zeroes using a^2 - b^2 identity.

Problem. Find the zeroes of the polynomial $x^2 - 3$ and verify the relationship between the zeroes and the coefficients.

1 / 6

reading

Verify It Yourself

Faded example for verifying relations.

0 of 3 blanks filled