For what values of a does y = ax^2 - 8x + 4 have no x-intercepts?

Question

Accepted Answer

Hey, everyone in this problem for the following quadratic function, we're asked to determine the values of A so that it has no X intercepts function we're given is Y is equal to A X squared minus 18, X plus 13. We're given four answer choices. Option A, the interval 81 divided by 13 to positive infinity. Where we don't include the end points. Option B the interval from negative 81 divided by 13 to positive infinity where we include the end point negative divided by 13. Option C is the same as option A but we include the end 0.81 divided by 13. And option D is the same as answer choice B but we don't include the end points. Now we have a quadratic function. If we imagine trying to find the X intercepts. One thing we could do is use a quadratic formula. OK. And the quadratic formula recall is given by X equals negative B P plus or minus the square root of B squared minus four AC all divided by two A. Now in our function A is gonna be the coefficient corresponding with the X squared term which is just A B is the coefficient corresponding with the X term which is negative 18 and C is the constant term which is 30. So we have our A B and C. Now, if we look at the function for X intercepts, how can we end up with no solutions? No X intercepts? OK. Well, the place where this is gonna happen is in this square root. OK. If B squared minus four AC is less than zero, then you're gonna be taking the square root of a negative number which gives you no real values. And so you're not going to have any X intercepts. OK? And recall that this B squared minus four AC is called the discriminate. OK? If the discriminate is less than zero, we get no X intercepts. And this is why from our quadratic format. So in our case, B squared is negative 18. So we have negative 18 squared. Sorry, I forgot to write that we have no X intercepts in this case. All right. So we have negative 18 squared minus four multiplied by a multiplied by 13. And we want this to be less than zero so that we have no X intercepts negative 18 squared gives us 324 and we have minus 52 A is less than zero. We're gonna add 52 A to both sides to move it over. We get that 324 is the less than 52 egg and we divide both sides by 52. I'm gonna rearrange the order here. So I'm gonna write a on the left hand side and the sign is gonna switch just because I've moved it to the other side. So we have A is greater than divided by 52. We can simplify this fraction. The numerator and denominator are both divisible by four. So if we divide both by four, we get that A needs to be greater than 81 divided by 13. OK. So if our discriminate B squared minus four AC is less than zero, then our quadratic formula tries to take the square root of a negative number. This gives no solution. So we have no X intercepts. OK? We found that this happens when A is greater than 81 divided by 13. And so we can write our interval as 81 divided by to positive infinity. And we're not including the end points. OK? We have a strictly greater than this value. We do not have a greater than, or equal to. And so we use round brackets to indicate that we're not including that end value. OK? Comparing this to our answer choices, we see that this corresponds with answer choice A the interval 81 divided by 13 to infinity, not including the end points. Thanks everyone for watching. I hope this video helped see you in the next one.

For what values of a does y = ax^2 - 8x + 4 have no x-intercepts?

Key Concepts

Quadratic Functions

Discriminant

X-Intercepts

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