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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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Is the transformation possible?

I came across this problem which asks to transform $x^2-x-2$ to $x^2-x-1$ ,if possible, using the following rules: Given a quadratic equation $ax^2+bx+c$ you can : 1)Interchange $a$ and $c$ 2)Replace $x$ by $x+t$ where $t$ is a real number. My…
LM2357
  • 4,083
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Why is it called a quadratic if it only contains $x^2$, not $x^4$?

Why is it called a quadratic if it only contains $x^2$, not $x^4$? Should it not be bidratic or didratic etc?
Parth Bhatnagar
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A question on roots of a Quadratic.

If roots of the equation ${ax}^2+bx+c=0$ are of the form $$\frac{\alpha}{\alpha-1},\frac{\alpha+1}{\alpha}$$ Then the value of $$({a+b+c})^2$$ I have no clue how to approach this one, any help is appreciated!
user385287
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1 answer

If 'x' is real then $\frac{x^2-x+c}{x^2+x+2c}$ can take all real values if?

the question asks for the interval in which c lies so that $\frac{x^2-x+c}{x^2+x+2c}$ gives all real values for all x belongs to R. how to proceed in this problem?
danny
  • 357
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Quadratic equation within quadratic

Edited version:- If both roots of equation $$4x^2 -20px+(25p^2 +15p- 66)=0 $$ are less than $2$, find the value of $p$. How to solve such type of equations?
Muzamil
  • 41
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5 answers

Finding range of $m$ in $x^2+mx+6$.

Find the range of values of $m$ in the quadratic equation $x^2+mx+6=0$ such that both the roots of the equation $\alpha,\beta<1$. My attempt - it is given that $\alpha<1$ and $\beta<1$ $\rightarrow \alpha+\beta<2$ But $\alpha+\beta=-m$ Thus…
Snehil Sinha
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The sum of the cubes of the reciprocal values of the roots of the equation $x^2+ax+1=0$ is?

The equation : $$x^2+ax+1=0$$ $a$ is a real number. How to find the sum of cubes of the reciprocal of roots for this equation. I tried solving this just by brute forcing it, but I get expressions that are ugly and pretty sure would yield nothing in…
John Doe
  • 1,080
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Quadratic formula not working?

Suppose I want to find the $ n $ for which $$(n)(n+1)/2 = 10 \Longrightarrow n^2 + n - 20 = 0$$ Clearly a solution is $4$. But, suppose we wanted to find that solution by use of quadratic formula. We get $$4 = \frac {-1 \pm \sqrt{1-80}}{2} $$ But…
user198428
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3 answers

Quadratic equations: Why does factoring by grouping work?

We are learning factoring by grouping - The teacher explained the process but didn't explain the logic behind it. You need to multiply the coefficient on the x-squared term by the constant to get a number. You then need to find two numbers which…
Little Black Dog
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is the sum of roots of a quadratic with rational coefficients always rational

quadratic is $ax^2 + bx +c = 0$ let the roots be $f$ and $g$ as $f + g = -\frac{b}{a}\ $ and $\ f \cdot g = \frac{c}{a}$ does this imply if a quadratic has rational coefficients the sum of the roots and the product of the roots are also rational?
zebra1729
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Algebraic Relationships - Quadratic Equations

I am having a tough time with the following question: If $x$ is real and $p=3(x^2 + 1)/(2x-1)$, then prove that $p^2 - 3(p+1)\geq 0$. I don't know how to tackle this question. Thanks for your help! :)
Bob
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Why are roots (x-intercept) called the solutions of a quadratic equation?

As I understand it, all the points in the hyperbole are the solutions. Although I see that it's easier to draw the curve after you discovered the roots, I don't see why to call them 'the solutions.' It has infinite solutions.
Quora Feans
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Sum of real roots of the equation $x^2 + 5|x| + 6 = 0$?

Sum of real roots of the equation $x^2 + 5|x| +6 = 0$
Deiknymi
  • 383
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If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist?

If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist? One Two Three Infinitely many Also, what are these pairs?
Shaurya Gupta
  • 4,339
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Proper method for solving quadratic equations with exponents

$(\sqrt {x^2-5x+6}+\sqrt{x^2-5x+4})^{x/2}$ + $(\sqrt {x^2-5x+6}-\sqrt{x^2-5x+4})^{x/2}$ = $2^{(x+4)/4}$ I have found out, by trial and error method, that $x=0$ and $x=4$ satisfy this equation. But is there a proper way to solve this equation and get…
Tejas
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