To solve the problem, let's start with the given equation:
[tex]\[
\sqrt{ab} = 10
\][/tex]
First, we square both sides of the equation to eliminate the square root:
[tex]\[
ab = 100
\][/tex]
Next, we are asked to determine which of the given values of [tex]\(a + b\)[/tex] cannot exist. The values provided for [tex]\(a + b\)[/tex] are 29, 20, 25, and 19.
Consider the quadratic equation with roots [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[
x^2 - (a+b)x + ab = 0
\][/tex]
Substituting [tex]\(ab = 100\)[/tex] into the equation, we get:
[tex]\[
x^2 - (a + b)x + 100 = 0
\][/tex]
To determine if a quadratic equation has real roots, we use the discriminant ([tex]\(\Delta\)[/tex]), given by:
[tex]\[
\Delta = (a + b)^2 - 4ab
\][/tex]
Substituting [tex]\(ab = 100\)[/tex] into the discriminant formula, we have:
[tex]\[
\Delta = (a + b)^2 - 4 \times 100
\][/tex]
[tex]\[
\Delta = (a + b)^2 - 400
\][/tex]
We need the discriminant to be non-negative for the quadratic equation to have real roots. Hence:
[tex]\[
(a + b)^2 - 400 \geq 0
\][/tex]
[tex]\[
(a + b)^2 \geq 400
\][/tex]
[tex]\[
|a + b| \geq 20
\][/tex]
We now check each given value of [tex]\(a + b\)[/tex]:
1. For [tex]\(a + b = 29\)[/tex]:
[tex]\[
|29| = 29 \geq 20
\][/tex]
The discriminant is non-negative, so 29 is a possible value.
2. For [tex]\(a + b = 20\)[/tex]:
[tex]\[
|20| = 20 \geq 20
\][/tex]
The discriminant is non-negative, so 20 is a possible value.
3. For [tex]\(a + b = 25\)[/tex]:
[tex]\[
|25| = 25 \geq 20
\][/tex]
The discriminant is non-negative, so 25 is a possible value.
4. For [tex]\(a + b = 19\)[/tex]:
[tex]\[
|19| = 19 < 20
\][/tex]
The discriminant is negative, so 19 is not a possible value.
Therefore, the value of [tex]\(a + b\)[/tex] that cannot exist is:
[tex]\[
\boxed{19}
\][/tex]
