Mathematics Stack Exchange
Mathematics Stack Exchange

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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Purely imaginary equation $p(x)=0$ with real coefficient

The quadratic equation $p(x)=0$ with real coefficient has purely imaginary roots. Then the equation $p(p(x))=0$ has (A) only purely imaginary roots (B) all real roots (C) two real and two purely imaginary roots (D) neither real nor purely imaginary…
Samar Imam Zaidi
  • 8,800
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4 answers

How to factorize quadratic equations quickly?

It takes me more than a minute to quickly factorise this kind of quadratic expression. $$3n^2 -53n + 232$$ I need to solve them in less than $10$-$15$ seconds. Please tell me a way I can solve them.
user821898
2
votes
3 answers

Revenue and quadratic formula - for every x increase in price there are y fewer sales

"When a shoe costs $\$80.00$, there are $300$ sales. Every $\$5.00$ increase in price will result in 10 fewer sales. Find the price that will maximize income." I am able to solve the question just fine, but I am confused about the logic in setting…
RudyGoburt
  • 295
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3 answers

Finding sign of leading coefficient of a quadratic equation

In a given quadratic equation $f(x)=ax^2+bx+c$ if $f(-1)>-4, f(1)<0$ and $f(3)>5$, then how can I find the sign of $a$? Answer in the textbook: $a>0$
user
  • 197
2
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1 answer

Proving relations of quadratic equation with roots in some sequence

Consider the questions: " if a,b,c are in G.P., then equations$ ax^2 +2bx+c = 0$ and $dx^2 + 2ex +f =0$ have a common roof if$ \frac{d}{a}, \frac{e}{b}, \frac{f}{c} $are in what sequence ( am, gm, hm etc) " " if $ \alpha, \beta$ are roots of…
tryst with freedom
  • 11,538
2
votes
2 answers

Value of $c$ in $x^2-\sqrt2x+c=0$

If the roots $\alpha$ and $\beta$ of the equation, $x^2-\sqrt2x+c=0$ are complex for some real numbers $c\ne 1$ and $|\frac{\alpha-\beta}{1-\alpha\beta}|=1$ then a value of $c$ is Squaring both sides, I get…
aarbee
  • 8,246
2
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2 answers

Vieta's formulas for quadratic equation problem

I'm using one hack, which I never though of why it works. But now I'm curious why it's works and how I can prove it. Here's the deal: we have quadratic equation $ax^2 + bx + c = 0$, to find roots I just multiply $c$ by $a$ and solving $y^2 + by + ca…
Zekfad
  • 25
2
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3 answers

Prove this has real roots $(a^2-bc)x^2+(a+b)(a-c)x+a(b-c)=0$

Prove this has real roots $(a^2-bc)x^2+(a+b)(a-c)x+a(b-c)=0$ My Work \begin{align*} \Delta&=(a+b)^2(a-c)^2-4a(b-c)(a^2-bc) \\ &=a^4+2a^3c+a^2c^2-2a^3b+b^2a^2-4a^2bc-2abc^2+2ab^2c+b^2c^2. \end{align*} How do I show that this is positive?…
emil
  • 1,310
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0 answers

determine the range of p to have distinct real roots for the following equation; $x^2+(p-q)x+(1-p-q)=0$

$\forall q$ determine the range of p to have distinct real roots for the following equation; $$x^2+(p-q)x+(1-p-q)=0$$ My Try $\Delta\geq0$ $(p-q)^2-4(1-p-q)\geq0$ I know I have to isolate p in this inequality to get a range for p in terms of q.…
emil
  • 1,310
2
votes
1 answer

Find the no of digits in the sum of all integral values of a in $[1,100]$ for which following condition satisfies.

Find the no of digits in the sum of all integral values of a in $[1,100]$ for which the equation $x^2-\left(a-5\right)x+\left(a-\dfrac{15}{4}\right)=0$ has atleast one root greater than zero. My attempt is as…
user3290550
  • 3,452
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votes
2 answers

Can any number of the form $4k+2$ be written as $a^2+b^2-c^2-d^2$?

Can any number of the form $4k+2$ be written as $a^2+b^2-c^2-d^2$?
kiva2000
  • 71
2
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5 answers

How to solve this quadratic equation (with $x$ represented by a fraction containing a square root)?

If $x-\frac{4}{5} = \pm \frac{\sqrt{31}}{5}$, how can I find the values of a and b in the following equation? $$5x^2+ax+b=0?$$ I've tried substituting $(\frac{4}{5} \pm \frac{\sqrt{31}}{5})$ for $x$ and got totally confused there.
brilliant
  • 818
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votes
2 answers

Finding roots of equation, with sole parameter.

Find the value of $k$ if product of two roots of equation $$x^4 -37x^3+kx^2 -808x -1984 =0$$ is 62 by using Vieta theorem i can get product of other two root as 32. But what to do after that?
maveric
  • 2,168
2
votes
4 answers

Real values of $k$ such that $kx^2+(k+3)x+k-3=0$ has $2$ positive integer roots

There is only one real values of $k$ for which the quadratic equation $kx^2+(k+3)x+k-3=0$ has $2$ positive integer roots. Then the product of these two solutions is What i tried. $$kx^2+(k+3)x+k-3=0$$ for $k=0,$ i am getting $x=1$(only one integer…
jacky
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