- #1

chwala

Gold Member

- 2,697

- 360

- Homework Statement
- Show that ##(x-a)(x-b)=b^2## has real roots

- Relevant Equations
- discriminant

If we have a quadratic equation, ##px^2+qx+d## then it follows that for real roots; The discriminant

## D= q^2-4pd≥0## therefore on expanding ##(x-a)(x-b)=b^2## we get,

##x^2-bx-ax+ab-b^2=0##

##a^2+2ab+b^2-4ab+4b^2≥0##

##a^2-2ab+b^2+4b^2≥0##,

##(a-b)^2+4b^2≥0##

since, ##(a-b)^2 ≥0## and ##4b^2≥0## is true for any value of ##a,b ∈ℝ## then our proof is complete. Thanks guys Bingo!

Is there another way of proving this?

## D= q^2-4pd≥0## therefore on expanding ##(x-a)(x-b)=b^2## we get,

##x^2-bx-ax+ab-b^2=0##

##a^2+2ab+b^2-4ab+4b^2≥0##

##a^2-2ab+b^2+4b^2≥0##,

##(a-b)^2+4b^2≥0##

since, ##(a-b)^2 ≥0## and ##4b^2≥0## is true for any value of ##a,b ∈ℝ## then our proof is complete. Thanks guys Bingo!

Is there another way of proving this?

Last edited: