Mind Your Decisions, I'm Presh Talwalkar.
Consider the equation Ax^2 - 5x + 2 = 0.
Solve for A, such that the equation has exactly one solution.
This is from the Finnish Matriculation Exam in
2014, which had an award of 1000€ for a perfect score.
Some students missed the award by missing this problem.
I saw this on r/math.
Pause if you'd like to give this problem a try, and when you're ready, keep watching to learn how to solve this problem.
We can solve a quadratic equation using
Brahmagupta's quadratic formula x = -b plus or minus the square root of (b^2 - 4Ac/2A).
Since we don't want to divide by 0, A ≠ 0.
This formula represents two different solutions. We'll separate them out into x1 and x2.
We'll get a single solution if the discriminant b squared minus 4Ac is equal to zero. In that
case, the discriminant will vanish, and the square root term will vanish,
so both of the solutions will be equal to  -b over 2A.
We'll use this lesson to solve our problem.
We'll substitute in the values and then we'll set b^2 - 4Ac = 0.
This simplifies to be 25 - 8A = 0.
Which we can easily solve that A = 25/8.
Many people thought this was the answer, but this is not the complete answer to the problem.
Remember, this is only true if A ≠ 0,
so we need to separately consider the case if A = 0. In
that case, we reduce this into a linear equation,
which, of course, has just one solution: x = 2/5.
So this is the complete solution: A = 25/8 or A = 0.
I also think it's fun to visualize this problem. Let's plot this equation for A = 2.
We get a parabola which has two x-intercepts, which are the two solutions.
As we varied the parameter A, this parabola will just become tangent to the x-axis and have one solution.
This occurs when b^2 - 4ac = 0 and we get A = 25/8.
Now if we varied this parameter A and let A = 0,
this parabola will eventually become a linear equation at A → 0.
And this will give us one solution.
The other solution will be when the discriminant is equal to zero and A = 25/8.
I think it's a very neat problem, and it really tests your understanding of quadratic equations.
Thanks for making Mind Your Decisions one of the best channels on YouTube, and as
always, thanks for watching, and thanks for your support.
