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Adrien said that if the roots of a quadratic equation are $\frac{1}{2}$ and $\frac{3}{4}$ , the equation is $4 x^{2}-5 x+\frac{3}{2}=0 .$ Olivia said that the equation is $8 x^{2}-10 x+3=0 .$ Who is correct? Justify your answer.

For $a=8$ the equation becomes$8 x^{2}-10 x+3=0$ (Olivia)and for $a=4$ it becomes$4 x^{2}-5 x+\frac{3}{2}=0$ (Adrien)

Algebra

Chapter 5

QUADRATIC FUNCTIONS AND COMPLEX NUMBERS

Section 7

Sum and Product of the Roots of a Quadratic Equation

Equations and Inequalities

Quadratic Functions

Complex Numbers

Polynomials

Campbell University

Harvey Mudd College

University of Michigan - Ann Arbor

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you have to. Students with two equations get to decide who is right. We're gonna use the fact that Route one puts route two equals B over a in sake, 1/2 less three force. Excuse me and say, Well, 1/2 is equal to to force meaning? If I added these, it would be 5/4 so immediately. I'm thinking that this student might be right, because if this equals negative B over a, I see a negative five for a B value here, and I see a four for in a value. So let's check if it works when I multiply these two. I have a 1/2 times of three. Fourth in this equals 3/8 which should be a C divided by a We'll see value for this student is equal to 3/2. If I divide that by the A value of four well, that's the same thing is saying, Why don't we just say multiply by 1/4? And if I multiply by 1/4 check it out. I do get that 3/8 that we want. And so, yes, for sure, this student is right because of these two things checking out But this student is also right. And here's why. If I take this equation and multiply everything by two, I get every single term. Here. Take a look. Two times for X where. Eight X squared two times negative five x negative 10 x into sensory over two is three Additionally, to times zero is zero. So, really, you can kind of say these two equations are exactly the same, with one having an a value equal to eight in another, having in a value equal to four. And so there's different values kind of just affected the terms differently, but really, these both are the same equation with the same roots of 1/2 and 3/4.

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