Video Transcript
For what value of 𝐾 does the equation three 𝑦 squared equals five 𝑦 minus 𝐾 have exactly one solution?
Well, what we’re going to do in this question is use something called the discriminant. And what we know about the discriminant is that if we have a quadratic in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐 equals zero, then the discriminant can help us tell what the roots of that quadratic equation are going to be. Because if we have the discriminant 𝑏 squared minus four 𝑎𝑐 less than zero, then there are no real roots. If 𝑏 squared minus four 𝑎𝑐 is equal to zero, then there is one repeated root. However, if 𝑏 squared minus four 𝑎𝑐 is greater than zero, there are two real and different roots.
So therefore, in our problem, what we’ll be using is the second scenario. And that’s because we’re looking for what value of 𝐾 does the equation have exactly one solution. So therefore, the first thing we want to do with our equation is rearrange so that we have it in the form 𝑎𝑥 squared plus 𝑏𝑥 plus 𝑐, or in our case 𝑎𝑦 squared plus 𝑏𝑦 plus 𝑐. And then that’s equal to zero. So to achieve that, what we’re going to do is subtract five 𝑦 and add 𝐾 to each side of the equation. And when we do that, what we will have is three 𝑦 squared minus five 𝑦 plus 𝐾 is equal to zero.
So now that we have it in the correct form, what we need to do to use our discriminant is to identify our 𝑎, 𝑏, and 𝑐. Well, our 𝑎 is going to be the coefficient of 𝑦 squared, so three. Our 𝑏 is going to be the coefficient of 𝑦, so negative five, remembering here don’t forget about the sign is going to be negative. And then our 𝑐 is going to be 𝐾. So therefore, with this information in mind, we can set up the equation negative five squared minus four multiplied by three multiplied by 𝐾 equals zero. And then simplifying, we get 25 minus 12𝐾 equals zero. And then adding 12𝐾 to each side, we get 25 equals 12𝐾. And then dividing by 12, we get 25 over 12 equals 𝐾. So therefore, what we can say is that the value of 𝐾 that the equation three 𝑦 squared equals five 𝑦 minus 𝐾 has exactly one solution is 25 over 12.
