### Instruction: The quadratic equation $ax^2+20x+c=0$ has exactly one solution. If $a+c=29$, and $a<c$ find the ordered pair $(a,c)$.### Response: To solve this problem, I need to use the fact that a quadratic equation has exactly one solution when its discriminant is zero.
The discriminant of a quadratic equation is the part under the square root in the quadratic formula: $b^2-4ac$.
In this case, the discriminant is $20^2-4ac$.
If I set this equal to zero, I get $400-4ac=0$.
Solving for $ac$, I get $ac=100$.
Now I have two equations involving $a$ and $c$: $ac=100$ and $a+c=29$.
I can try to eliminate one of the variables by substitution.
If I solve for $c$ in the second equation, I get $c=29-a$.
Then I can plug this into the first equation and get $a(29-a)=100$.
Expanding and rearranging, I get $a^2-29a+100=0$.
This is another quadratic equation, but I can factor it easily: $(a-4)(a-25)=0$.
So $a$ is either $4$ or $25$.
But I also know that $a<c$, so I can rule out $a=25$.
That means $a=4$ and $c=29-a=25$.
So the ordered pair $(a,c)$ is $(4,25)$.