Would this be sufficient as a proof?

sooyong94 · Jul 21, 2016

Homework Statement

Typed problem 17: If alpha and beta are roots of ax^2 + 2bx + c = 0, show that alpha + beta = -2b/a and alpha beta = c/a. If the equation has real roots and m, n are real with m^2 > n > 0, show ax^2 + 2mbx + nc = 0 also has real solutions.

2. Homework Equations
Vieta's formula, quadratic discriminant

3. The Attempt at a Solution

Handwritten solution showing Vieta's formulas for roots alpha and beta of a quadratic, with sum and product expressions, followed by algebra expanding (alpha squared plus beta squared) using the sum and product of roots.

Handwritten derivation of Vieta's formulas for a quadratic: ax² + 2mbx + nc = 0, sum of roots = -2mb/a = m(-2b/a) = m(α + β), product of roots = nc/a = n(αβ).

Handwritten proof about the roots of a quadratic: a = -b^2 - 4ac/a and b^2 - 4ac is greater than or equal to 0, with a note that the discriminant is non-negative, and a comparison of x^2 + 2bx + c terms.

Handwritten derivation of the sum and product of roots of a quadratic: alpha and beta from the quadratic formula, then alpha plus beta = -4b/2a = -2b/a and alpha times beta = c/a.

Ssnow · Jul 22, 2016

suggestion for the first part: since ##\alpha,\beta## are solutions your equation as also a second form ##(x-\alpha)(x-\beta)=0## so you can expand this and proceed comparing with ##ax^{2}+2bx+c=0## ...
for the second part you must to look to the ##\Delta## of the equation ##ax^{2}+2mbx+nc=0## and to prove that is ##\geq 0## (observe that ##b^{2}-ac\geq 0## by assumption) ...

Your attempt at a solution is very long, I hope with my suggestions you will able to simplify ...

haruspex · Jul 25, 2016

I was a bit confused because the order of the solution attachments does not match the order of the questions. The last attachment appears to address the first question. As Ssnow writes, it is simpler to start with factorisation.

The first attached solution seems unrelated to the posted questions.

The second attached solution does seem to be related to the second posted question, but exactly how is unclear.

The third attached solution does address the second question, but the logic is backwards. You are asked to show that the equation has real roots, so you should not start with "if it has real roots". You could start with "it will only have real roots if..."

Would this be sufficient as a proof?

Homework Statement

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