1. ## The nature of the roots of a quadratic equation

Hi,
Thanks.

Show that the roots of the equation $\displaystyle x^2 - (a + d)x + (ad - b^2) = 0$ are real.

2. ## Re: The nature of the roots of a quadratic equation

Have you tried checking the discriminant of this quadratic equation?

3. ## Re: The nature of the roots of a quadratic equation

Yes Sir, I have. $\displaystyle b^2 - 4ac$ greater than or equal to zero;

$\displaystyle a = 1, b = a+d, c = ad - b^2$;

$\displaystyle b^2 = (a + d)^2 = a^2 + 2ad + d^2$;

$\displaystyle - 4ac = - 4(1)(ad - b^2) = - 4ad + 4b^2$;

However, how does one prove that $\displaystyle a^2 - 2ad + d^2 + 4b^2$ is greater than or equal to zero?

4. ## Re: The nature of the roots of a quadratic equation

Try factoring the first 3 terms as the square of a binomial...then you will have the sum of two squares...

5. ## Re: The nature of the roots of a quadratic equation

Originally Posted by Seaniboy
Show that the roots of the equation $\displaystyle x^2 - (a + d)x + (ad - b^2) = 0$ are real.
You are making it too hard.
$\displaystyle \\(a+d)^2-4(1)(ad-b^2)\\=a^2+2ad+d^2-4ad+4b^2\\=a^2-2ad+d^2+4b^2\\=(a-d)^2+4b^2$

6. ## Re: The nature of the roots of a quadratic equation

Thanks Plato.

Thanks Mark.