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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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Using Nature of Roots to find range of values

Find the range of values of $k$ for which $3x^2-4(k-x)+2$ is always positive for all real values of $x$. I've tried simplifying, until I got to: $3x^2-4k+4x+2$. Since it must always be positive, the discriminant, $b^2-4ac$ must be negative, ie…
Matt Chua
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Can you find the quadratic coefficient of quadratic equation?

How do you find the quadratic coefficient a, b, and c, given some values of x and y? For example: Suppose we have f(0.1524)=0.9961 f(0.8258)=0.0782 f(0.5383)=0.4427 and given the quadratic equation: $y = ax^2 + bx + c$, then can you find values of…
user777
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How can I use quadratic formula with an inequation like this $(x-3)(x^2-5x+5)\le0$

I know how to use the quadratic formula with an inequation. How can I make use the quadratic formula? $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ With an inequation like this $(x-3)(x^2-5x+5)\le0$
kitta
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quadratic equation - nature of roots

For what values of a does the equation $$x^2-( 2^a-1)x-3(4^{a-1}2^{a-2})=0$$ possess real roots? Since the roots are to be real that means the discriminant should be $\geq 0$ $$\Rightarrow (2^a-1)^2+4\cdot 3\cdot (4^{a-1}2^{a-2}) \geq 0$$
Sachin Sharmaa
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Proof that equation does not have real roots

For polynom $f(x)=ax^2+bx+c$ equation $f(x)=x$ has no real solutions. Prove that equation $ f(f(x))=x$ also does not have does not have real solutions Can someone explain solution to me? Why? If equation $f(x)=x$ has no real solutions than it…
Ivan Ljilja
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Find the value of $a$ such that $f(x)$ has exactly one root $\alpha$ in interval $(1,2)$ and.....

Question Find the value of $a$ such that equation $$f(x)=x^2+(a-3)x+a=0$$ has exactly one root $\alpha$ between the interval $(1,2)$ and $f(x+\alpha)=0$ has exactly one root between the interval $(0,1)$. Attempt Discriminant$=0$ for exactly one…
jayant98
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Find coefficients to a quadratic equation knowing roots and a point...

Given a standard quadratic equation: $$p(z) = az^2 + bz + c$$ We know that $-10$ and $10-i$ are roots. We know that $p(i)=-10$ What are $a$, $b$ and $c$?
Jan Jensen
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What are all of the negative integral solutions of $y^2+6xy-8x=0$?

I got the answer as $(0,0)$ by making $D\ge 0$ (quadratic in $y$). However, how do I know this is the only possible answer?
user459947
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theory of equations using transformations

i think this case is not possible. If you take $x$ as $5$, and -$5$, $P(27)$ will have two values. so i think the ans is D.
maveric
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Finding the relation between coefficients of quadratic equation

My attempt: I could prove $|c| < 1$. As it given that $f\lt 1$, so $f(0)\lt 1$. I have solved using triangular inequality. Is there any other way? ans is ABCD
maveric
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Square root equation problem $2\sqrt{2x}-\sqrt{x-1}-\sqrt{x+7}=0$

I was solving some equations and got to this one: $$2\sqrt{2x}-\sqrt{x-1}-\sqrt{x+7}=0$$ I tried square the equation, but it seems that it only complicates the process.
VLC
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Given that $m$ is a real number not less than $-1$

The question is: Given that $m$ is a real number not less than $-1$, such that the equation in $x$ is $x^2+2(m-2)x+m^2-3m+3=0$ has two distinct roots $r$ and $s$. If $r^2+s^2=6$, find the value of $m$. Here's what I've tried:. Using Vieta's…
Isaiah Leobrera
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Given that the quadratic equation $x^2-px+q=0$ has two roots $r$ and $s,$ Find the quadratic equation that takes $r^3$ and $s^3$ as its roots.

Here's what I've tried: Using Vieta's formulas: $\;rs = q\;$ and $\,r+s = p$. Then I cubed $rs$ which is $q^3$ and $(r+s)^3 + rs$ which is $p^3 + pq.$ Thinking that $x^2 - (p^3+pq)x + q^3$ is the answer. Did I do something wrong here?
Isaiah Leobrera
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Theory of Quadratic Equations

This an example problem in Hall&Knight from the chapter the theory of Quadratic equation Under Art 121. The question is, Find the Limits between which 'a' must lie in order that $$\dfrac{ax^2-7x+5}{5x^2-7x+a}$$ may be capable of all values,x…
Dravid Hemanth
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How does this equation yield into the following

$\frac{(a^2 + b^2)} {(ab + 1)} = k$ becomes $x^2-kb \cdot x + (b^2 - k) = 0$ in an example I am attempting to understand, can you please clarify how the quadratic equation follows from the equation above?
user527326
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