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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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If $ a(12a + 5b +2c) > 0 $ , then prove that the real roots of equation $ ax^2+ bx +c =0$ are less than 2

I know that to prove both roots less than 2 ,i have to prove $ 4a+ 2b +c > 0$ and $ -b/2a < 2 $ . Here i have no idea how to proceed.
raj
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How to solve specific parameters for a quadratic equation?

x^2+ax+a so that there are two different solutions x>5 First I set up that the discriminant is: D > 0 Then using Vieta's formula: a>25, a<10 But still, if I take 5 and 6 as solutions, I end up breaking my condition. How can I set up the conditions…
therue
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Determining the number of roots

Given a set of two equations (one linear and one quadratic in $x$ and $y$) as follows:- $$ax + by + c = 0 \tag 1$$ $$Ax^2 + Bxy +Cy^2 + Dx + Ey + F = 0 \tag 2$$ What are the conditions that can be imposed on the coefficients such that the solution…
Mick
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Linear-Equation - What is the error in this equation?

I want to know if this equation is wrong? \begin{align} 4(x - 3) + 2(x - 1) - 3 &< 10\\ 4x-12 + 2x-1-3 &< 10\\ 4x - 2x &< 10 + 12 + 1 + 3\\ 2x &< 26\\ x &< 26/2\\ x &< 13\\ \end{align} if it is wrong, what is the problem? i…
Splitzer
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Forming Quadratic equation from roots

I think I just need some background. I've got the following quadratic equation: $$ 1 - x - 2x^2 = (1-2x)(1 + x) $$ But if I solve it with the quadratic equation, I get the roots: $$ \frac{1 + \sqrt{9}}{-4} = -1, \frac{1-\sqrt{9}}{-4} = +1/2 $$ And,…
Chris
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From the graph of a quadratic equation, find the range of values of $x$.

The equation of the graph is $$ y = -x^2 + 9x - 18 $$ From the graph sketched, find the range of values of $x$ for which $x^2 + 18 > 9x$. Workings $$ y = -(x-3)(x-6) $$ I'm not sure what is the question asking ... Can I get a hint? Thanks for your…
user307640
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Quadratic equations - Fastest way to find the value of d

$\frac{2d^2-d-10}{d^2+7d+10} = \frac{d^2-4d+3}{d^2+2d-15}$ What is the optimal solution for finding the value of d?
Juanvulcano
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If you are using piecewise quadratic polynomials to approximate the function $f (x) = \ln x$ on the interval $[1, 2]$

If you are using piecewise quadratic polynomials to approximate the function $f (x) = \ln x$ on the interval $[1, 2]$ and expect the maximum error to be smaller than $10^{-6}$, how many subintervals do you need?
nagu
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Factorizing quadratics mentally

How would it be possible to factorize quadratics mentally, for example the following one? $$2x^2+7x+3$$ Maybe even something like $$3x^2+22x+24$$
Skeleton Bow
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I need help with a radicals questions

The roots of the equation $ax^2+bx+c=0$ are in the ratio of $2:3$. Determine an expression for $b$ in the terms of $a$, and $c$. I need help solving the question, please help, and thanks!
Chief123
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proving theorems about quadratic function

The theorems I want to prove are: $$\text {if } a + b + c = 0 \text{ then } x_1 = 1 \text { and } x_2= c/a$$ $$ \text{if } a - b + c = 0 \text{ then } x_1 = - 1 \text{ and } x_2 = -c/a $$ where $x_1$ and $x_2$ are the roots of the equation and…
Planet_Earth
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If $y=-x^2+2x+4$ intersects $y=mx$ at $P$ & $ Q$. Determine the value $m$ so the mid point of $P$ & $Q$ is origin

If $y=-x^2+2x+4$ intersects $y=mx$ at Point $P$ and Point $Q$. Determine the value m so the midpoint of P and Q is the origin. I solved this question when I graphed it out, I wonder if there is a way to solve the problem without graphing? Thank…
MarkCP
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Solving one variable in terms of the another

Let $y = x^2 - 2x + 6$. Express $x$ in terms of $y$. This is my working: $$ x^2 - 2x = y - 6, \\ x(x-2)= y - 6. $$ From this point, I got stuck as I can't fully factorize the $x$ out as seen above... Thanks for the hint, if any...
user307640
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speed and travel problems

The ship left the harbor and is travelling $60$ kilometers downstream. Then it continues on the tributary river (upstream) $20$ kilometers. Travel took $7$ hours to finish. River speed is $1$ km/h. Calculate the speed of a ship. $v$ is speed of the…
user309607
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How to derive in a quadratic equation.

I was reading Purplemath's lesson about quadratic equations, and came to the part about deriving the solution to $ x^2 + 6x + 10 = 0$. I understood the part about putting the loose number in the other side and taking half of the coefficient on the…
ezekiel
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