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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 $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

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Find all values of $k$ for $kx^2+(k+2)x-3=0$ with positive roots.

$$kx^2+(k+2)x-3=0$$ This quadratic has roots which are real and positive. Find all possible values of $k$. I had already tried using the discriminant and reached this point $$ Δ = (k+8)^2-60 $$ $$ ==> (k+8)^2-60>0 $$ $$ k>2\sqrt{15}\ - 8 $$ and $$…
Ozymandias
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Relating various given equations

If $2(a+b+c)=t^2+u^2+v^2$ and the roots of $x^2+tx-a=0$ are $u,v$ and the roots of equation $x^2+ux-b=0$ are $v,t$ then I need to show that the equation whose roots are $t,u$ is $x^2+vx-c=0$ I am not able to guess any approach so sorry. Please do…
Akash Roy
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Find a quadratic equation with root $x_1^2$ and $x_2^2$

I'm struggling to solve this problem: "If the equation $ax^2 + bx +c$ $(a \neq 0)$ admits real and not null roots of $x_1$ and $x_2$, obtain the equation which evaluates to roots $(x_1)^2$ and $(x_2)^2$" I know the answer is $a^2x^2 -(b^2 - 2ac)x +…
thiagohubes
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If $x^2-bx+c=0$ has real roots, then prove that both are greater than $1$ when $c+1>b>2$.

If $ x^2-bx+c=0$ has real roots, prove that both roots are greater than $1$, when $c+1>b>2$. Working I tried to prove the given inequality by taking roots greater than $1$. Let $\alpha$, $\beta$ be the roots of the quadratic equation. So…
emil
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Common Root in Quadratic Equation

Let $P,Q,R$ be positive real numbers, not all equal, such that some two of the below equation have exactly one common root, alpha. Then prove that alpha is real and negative and one of the below mentioned equation has imaginary roots Equations:-…
Rohit Sinha
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Negative solutions to positive values

Wondering what negative solutions to equations represent in real life. For example, I was solving a problem that asked for the time when the velocity of a particle is $75\frac{m}{s}$, and I got roughly $6$ and $-11$. What does the $-11$ solution…
Yusuf Abukar
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How to find value of $A'$ in eliminating cross product terms Quadratic Curve Rotation?

I was studying conics and came around the topic of eliminating cross-product terms when rotating coordinates of a quadratic curve of the form $$A x^2 + B x y + C y^2 + D x + E y + F = 0$$ where $$\begin{align*} A x^2 &= A\left(\cos(\alpha) x' -…
Syed Ahmed Jamil
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Quadratic equation formula help / simplification

I have this quadratic equation, $ x^{2} + \frac{10}{3}x -\frac{80}{3} = 0 $ I use the quadratic formula to solve and simplify $-10 \pm \frac{\sqrt{100-(4)(3)(-80)}}{6}$ = $ \frac{-10 \pm \sqrt{1060}}{6}$ my book says it should simplify to $…
italy
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Dealing with negative roots in code

I have an equation in the following form that needs to be solved for $x$, with $x > 0$ and constants $a,b,c,d \geq 0$. $\left(\frac{x - a}{b}\right)^2 + \left(\frac{x - c}{d}\right)^2 = 1$ From the following paper, which i am trying to…
Aedoro
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Question on the inequalities of a parabola function

I have a question here from a textbook. Show that $x^2 +2kx +9 \ge 0$ for all real values of $x$ if $k^2 \le 9$ Here's my proof: I found values of $k$ to be between $ -3\le k\le3 $ For all real values of $x$, $b^2 -4ac \le 0 \Rightarrow $ no real…
zam
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A quadratic equation with an unknown parameter

I have a simple formula$$ 3x^2+kx+7=0.$$ I know that the discriminant of the function where the value of $k$ needs to be $ -2{\sqrt21} < k $ I want to find values of $k$ in which the function has two real roots or one real root. I used the formula…
zam
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Quadratic function problem

Find the coordinate of point A in the figure Can you solve it in way which doesn't involve derivation?
Marva Jami
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Finding the Sum of $a,b,c,d$ that satisfy the following conditions

Let $a,b,c,d$ represent $4$ different non-zero integers such that the absolute value of each integer is less than $11$. If $c$ and $d$ are the solutions for $x$ of $x^2+ax+b=0$ and if $a$ and $b$ are the solutions for $x$ of $2x^2-cx-20d=0$, find…
user3753
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values of $p$ that ensure quadratic formula has real solutions for $x$

So using the quadratic formula to give the solution for $x$ (in terms of $p$) for: $$px^2+2x+1=0$$ which as far as i've figured out gives: $$x = \frac{-2\pm\sqrt{4-4p}}{2p}$$ but i don't know how to find the values of $p$ that make this equation…
C.Cam
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What is the number of integral values of $k$ for which the equation $|x^2 - 5|x| + 6|=k$ has four solutions is?

I have done the sum by first plotting the graph of the function in the Left Hand Side of the equation and then plotted the line $y=k$. For the equation to have $4$ solutions, both these two curves must intersect at $4$ different points, and from the…
Rajdeep Dutta
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