A quadratic equation ƒ(x) = 0 has a solution x = 2. Its graph has...

Question

A quadratic equation ƒ(x) = 0 has a solution x = 2. Its graph has vertex (5, 3). 
What is the other solution of the equation?

Accepted Answer

Hey, everyone. Welcome back in this problem. We're asked to find the other zero of the quadratic equation. If one of the zeros is four, given that the vertex of the function is eight comma four, we're given four answer choices. Option A 12, option B negative 12, option C negative four and option D 60. So let's start with what we're given. And we want to write this quadratic equation so that we can find the zero. But what are we given? That's gonna help us write this equation. And in this case, that's the vertex. OK. So we wanna write the quadratic equation in vertex form and recall that vertex form is Y is equal to a multiplied by X minus H squared plus K or the vertex is located at H comma K. So writing the quadratic equation in this format gives us a direct link to the coordinates of the vertex. Now, in this case, our vertex is eight comma four. And so our H value is going to be equal to eight and our K value is going to be equal to four. That means that our equation is gonna be Y is equal to a multiplied by X minus eight squared plus four. All right. So we've gotten a lot of information about our equation here, but we still don't know this value of A, a without that value of A, we can't find this other zero. What we do know though is that one of the zeros is four? OK. So we have zero four. I recall that the zeros are going to be the X values when the Y value is equal to zero. OK. So that means that we have a 0.4 comma zero as part of this equation. So if our equation satisfies this 0.4 comma zero, we can substitute that in substituting this point N Y is equal to zero. That's gonna be equal to a multiplied by X which is four minus eight squared who was for. And now you can see why we substituted this point in the only unknown we have now is this value of A. So we can solve for a, that's gonna give us our complete quadratic equation. And then we can find that second zero. So let's simplify, we have zero is equal to a multiplied by negative four squared plus four. Simplifying some more. We're gonna move the four to the left hand side by subtracting it. You get negative four is equal to 16. A maybe a negative four square which gives us positive of 16. And finally, we can divide by 16 to get that A is equal to negative one quarter, we get negative four divided by 16 and we can simplify. So now our equation, it gonna be Y is equal to negative a quarter multiplied by X minus eight squared plus four. We want to find the other zero. Hm So the zero again, we mentioned this before when we were writing up the point that we were given from that zero, we have a zero when Y is equal to zero. So we set Y is equal to zero. In our equation, we get zero is equal to negative a quarter multiplied by X minus eight all squared plus four. And we wanna go ahead and solve for X. We're gonna start by moving our four to the opposite side. Negative four is equal to negative a quarter multiplied by X minus eight all squared. Gonna multiply both sides by negative four that gives us 16 is equal to X minus eight squared. We can take the square room taking the square root on the left hand side, the square root of 16 is four. Remember that we get both the positive and the negative root. So we have plus or minus four is equal to X minus eight. And so we're gonna have two solutions. We get that X is gonna be equal to four. Who was it? Or X is gonna be equal to negative four plus eight. And simplifying each of these tells us that X is equal to 12. Four, X is equal to four. So we've written out our equation using the information we've given. And we've just found the two zeros and the two zeros we've found is X equals 12 and X equals four. Now X equals four is the first zero. That's the zero we were given in the problem. So this is great. This is a great double check of our work because we get that zero back if that we were expecting. And so the answer to this question, what's gonna be the other zero? And that other zero we found is X is equal to 12. If we go up to our answer choices, we can see that this corresponds with option A. Thanks everyone for watching. I hope this video helped see you in the next one.

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A quadratic equation ƒ(x) = 0 has a solution x = 2. Its graph has vertex (5, 3). What is the other solution of the equation?