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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How do we get $d$ in terms of $a$ here: $6(d – a)^2 = ad$

I am very sure there is a specific way of solving the equation, $$6(d – a)^2 = ad$$ in order to get $d$ in terms of $a$.
user809332
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Finding the value of a variable in a quadratic

Question: The number of negative integral values of $m$ for which the expression $x^2+2(m-1)x+m+5$ is positive $\forall$ $x>1$ is? For me, solving this question if the parameter "$\forall$ $x>1$" was not given would be quite easy. But how do I…
General Kenobi
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Location of roots of a Quadratic Equation

Question: For what values of $m\in\mathbb R$, the equation $2x^2-2(2m+1)x+m(m+1)=0$ has exactly one root in the interval $(2,3)$? My Approach: As the leading coefficient of the equation is positive, its graph would be an upwards opening parabola.…
General Kenobi
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the equation $x^2-y^2 =a^2$ changes to the form $xy=c^2$ if the co-ordinate axes rotates through an angle (keeping origin fixed)

the equation $x^2-y^2 =a^2$ changes to the form $xy=c^2$ if the co-ordinate axes rotates through an angle (keeping origins fixed) is a) $ \frac \pi 2 $ b) $ - \frac \pi 2 $ c) $ \frac \pi 4 $ d) $ \frac \pi 3 $
sojado2736
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Pair of linear equation in two variables

This is from a text book:- "The general form of a linear equation in two variables is $ax + by + c = 0$ or, $ax + by = c$ where $a, b, c$ are real numbers such that $a ≠ 0$, $b≠0$ and $x, y$ are variables. (we often denote the condition $a$ and $b$…
Shekhar Malhotra
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Formula for first difference is not the derivative?

I was messing around in Desmos and wanted to create a chart to show the x values, function values, first differences, and second difference of some quadratics. The graph I was using is here. However, I had to create explicit formulas for the first…
Nik3141
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Solve a quadratic function passing through 2 defined points

I'm trying to find the values a, b and c that would validate y = ax^2 + bx + c with the following parameters: For x = 1; y = 1 For x = T; y = S Essentially, I would like the function to pass through the (x, y) coordinates (1, 1) and (T, S) for…
Pierre
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what is the equation for the path of the water if the maximum height of the water must be 4 feet

We are only told that the maximum height is 4 feet for the water fountain so we know that the vertex is (0,4). We think we would use the formula y=a(x-h)^2-k, but we do not know what to put in for x or y.
Marcella Demulling
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Inequality based on location of roots

(a) Find all values of the parameter "k' for which the solution set of the inequation $x^{2}+3 k^{2}-1 \geq 2 k(2 x-1)$ is a subset of the solution set of the inequation $x^{2}-(2 x-1) k+k^{2} \geq 0$ (b) Find all values of k for which there is at…
Akash Gautama
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Condition for terms to lie in a geometric series

$ax^2+2bx+c=0$ and $px^2+2qx+r=0$ has a common root. If terms $\frac ap, \frac bq, \frac cr$ lie on consecutive arithmetic series, Prove that terms p, q, r lie on consecutive geometric series. My Try Since they have a common root I derived the…
emil
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Find the values of m that x^2 - 4mx +20 = 0: has a) no solutions and b) 2 solutions

I understand how to solve this problem, but my method/logic seems a round-about way of doing so. Any tips on how to solve this question faster/more easily would be appreciated :) Solving for part a) no solutions: Find the discriminant: 16m^2 -…
user780357
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Difficulty setting up quadratic equation from word problem

The initial word problem is: "A tennis ball is hit upward from an initial height of 4 ft with an initial velocity of 40 feet per second. a. How long after the ball is hit will it be 20 feet above the ground? b. How long after the ball is hit will it…
Duffy
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Finding range of expression based on conditions applied to quadratic expressions

The question original text is... Given that $a, b, c$ are distinct real numbers such that expressions $ax^2+bx+c, bx^2+cx+a$ and $cx^2+ax+b$ are always non-negative. Prove that the quantity ${a^2+b^2+c^2 \over ab+bc+ca}$ can never lie in $(-…
knoftrix
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Find all the values of $a$ for which both the root of the equation $(a-2)x^2 - 2ax + a = 0$ lies in the interval $(-2 , 1)$.

Here if we consider the above equation to be quadratic then i have got the solution that $a \in [0,8/9)$, but if we consider the above equation to be not quadratic i.e. $a=2$ , then it becomes a linear equation with solution $x = 1/2$ which lies in…
Sameer Nilkhan
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how to prove that if a quartic equation ( with real coefficients ) has 4 imaginary roots they all will be in conjugate pairs?

I proved this fact for qudratic equation in the following way , let a qudratic equation have a imaginary root p+iq(q is not 0) , and let other root be (a+ib). Now here sum of roots will be a real number lets say R , => p+iq + a+ib = R , =>(p+a) +…
Sameer Nilkhan
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