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Questions tagged [quadratics]

Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

The root of $y=ax^2+bx+c$ can be solved by the formula $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

5400 questions
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Can there be more solutions to following equation?

Consider the following equation for $t > 1$: $$(t + \sqrt{t^2 - 1})^{x^2 - 2x} + (t - \sqrt{t^2 - 1})^{x^2 - 2x} = 2t$$ If we let $u = t + \sqrt{t^2 - 1}$ then $\frac{1}{u} = t - \sqrt{t^2 - 1}$ and the equation reduces to $$u^{x^2 - 2x} +…
Shiv
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If I have an x intercept and a y intercept, how might I find the vertex of a parabola?

I have looked all over, and I have found different things here and there for figuring pretty much everything but that. What I have is an x chord and a y chord, but I need to find the vertex. This isn't for school or anything, it is for an…
texasman1979
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$\sqrt{\Delta} = $ the distance between the midpoint and the two roots

I understand how the midpoint of a parabola is found when one of the roots is $0$ ( it is just $-b/2a$ ) but I can't understand why $\sqrt{\Delta} $ in the quadratic formula $=$ the distance between any of our real roots and our midpoint. I want to…
Shadow sparkle
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Factor this quadratic expression

I need to do the following: Prove that a quadratic expression of the form $A(x^2-y^2) - (B-C)xy$ can be always factored into two linear factors. It is easy enough to compare this with the standard representation $ax^2 + 2hxy + by^2 + 2gx + 2fy +…
ankush981
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If the roots of the quadratic equation $2kx^{2}+(4k-1)x+2k-3=0$ are rational and k is an integer, how many values can k take which are less that 50?

If the roots of the quadratic equation $2kx^{2}+(4k-1)x+2k-3=0$ are rational and k is an integer, how many values can k take which are less that 50 ? The discriminant = $16k+1$ For a rational number the discriminant = $p^{2}$ i.e.…
HOLYBIBLETHE
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What will be the value of k for which equation $x^2–4|x|+3=|k−1|$ has four real roots?

The equation is $$x^2–4|x|+3=|k−1|$$ There are several ways to find the solution using either graph or analytically. I want to know is how to do the graphical solution free hand without a calculator. Also in the analytical method I am unable to…
SAGNIK UPADHYAY
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Help with this solution of quadratic equation

In one of the solved examples in the book I'm following, the following expression arises after considering $D > 0$ for a certain equation: $$D = (n+1)^2p^2 - 2pq(n^2+1) + (n-1)^2q^2$$ From this, the following step was…
ankush981
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Let $x^2 + 2y^2 – 2xy – 2 \ge k (x + 2y) \;\forall\; x, y \in \mathbb{R}$ then find the number of integral values of k.

So I am stuck on a problem of a quadratic module. Let $x^2 + 2y^2 – 2xy – 2 \ge k (x + 2y) \;\forall\; x, y \in \mathbb{R}$ then find the number of integral values of k. Since it it's the part of quadratics module and the calculus part is introduced…
Techno Freak
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Show that that if $p,q,r,s$ are real numbers and $pr=2(q+s)$, then at least one of the eqns $x^2+px+q=0$ and $x^2+rx+s=0$ has real roots.

Show that that if $p,q,r,s$ are real numbers and $pr=2(q+s)$, then at least one of the eqns $x^2+px+q=0$ and $x^2+rx+s=0$ has real roots. My Attempt to the solution we know to have a real solution d>=0 so either 1) $p^2-4q>=0$ or 2) $r^2-4s>=0$ or…
maths lover
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Value of an algebric expression in a quadratic equation

I came across such a problem: Given the equation \begin{equation} x^2 + \sqrt{m} x + n = 0 .\tag{1} \end{equation} If it has two equal real roots, what is the value of $(m+1)(m-1) - 2(2n - 1)$? This is what I have done: Since the quadratic…
Shen
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Using the discriminant to find the value of k.

Question:- $x^2-4x-1=2k(x-5)$ has two equal roots. Calculate the possible values of $k$. I know that that must mean the discriminant must equal $0$. So I found: $b = (-2k-4)$ $a = 1$ $c = (10k-1).$ Yet when I input this into $b^2 - 4ac$ I always get…
tacticalcamel
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Find interval of $c$ such that $2e^{2x} -(c+1)e^x +2 \ge 0$ for all $x\in R$

Now I know the normal method of manipulation which will get us $$c+1\le 2(e^x + \frac{1}{e^x})$$ ie. $c\le 3$ But can I do it by assume $e^x=t$ and then resolving the quadratic? What complications would $t>0$ bring into it? I realise that setting…
Aditya
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Quadratic Functions: Does anyone know how to solve this?

I'm taking a grade $11$ summer school math course right now and I'm having trouble with the functions unit that we're on. I've been having a hard time understanding this question. I'm pretty bad at math so all of these math terms aren't really…
Delly Seen
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How to think while solving question like this? If $ax^2+bx+c=0$ has roots $A$ and $A^n$, then $(ac^n)^{1/(n+1)}+(a^nc)^{1/(n+1)}+b=0$.

I want to know what should be your thinking while solving this question. $ax^2 + bx + c = 0$ has roots $A$ and $A^n$. Prove that $$(ac^n)^{1/(n+1)} + (a^nc)^{1/(n+1)} +b = 0$$ What are you thinking in your mind about the variables, about the…
Rider
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Maximum of d(12-d)

I'm a little confused on a quite simple quadratic problem. I need to calculate the maximum of $d(12-d)$ using basic quadratics. The answer is $6$ as can also be shown by $f'(x)= -2d +12$, however this is algebra not calculus. Using the squares…
Sam
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