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 $y=a(x+b)(x+c)$.

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4
votes
1answer
61 views

The maximum value of the smaller root of given quadratic function

Consider the quadratic expression : $$ f(x) = x^2 +(a+2)x + (a^2 - a +2 ) $$ given is that $a , p, q , (p<q)$ are real numbers and p and q are the roots of the equation $f(x)=0$. Q1) find the ...
0
votes
1answer
48 views

Is there a way to solve a fourth grade equation if I know with which equations I got it?

So I have the multiplication of two quadratic equations that give me equal to a linear one, all with the same common variable. Am just trying to solve for that variable, but doing so with the ...
1
vote
0answers
35 views

Quadratic convergence in numerical method

Let $$x_{n+1}= \dfrac{1}{2}\left ( x_n+\dfrac{R}{x_n} \right )$$ Interpret this relation in terms of quadratic convergence. State your condition for such quadratic convergence clearly. Prove that ...
5
votes
1answer
65 views

Proving the existence of polynomial $p(x)$ and $q(x)$ such that $f(x)=1/2[p(x)+q(x)]$

Let $f(x)=x^2+ax+c$ where $a,c$ are real numbers. Prove that there exist quadratic polynomials $p(x)$ and $q(x)$ (with real coefficients) having all roots real such that $f(x)=\dfrac{1}{2}[p(x)+q(x)].$...
0
votes
2answers
31 views

Simultaneous equations and tangrnts with unknown value $k$ and tangent [closed]

The line with equation $2x+3y= k $ s a tangent to the circle $x^2+y^2+6x+4y=0.$ Find two possible values of $k$
-3
votes
4answers
61 views

Solve $\begin{cases}x^2=7x+3y\\y^2=7y+3x\end{cases}$ [closed]

Solve $$\begin{cases}x^2=7x+3y\\y^2=7y+3x\end{cases}$$ Can you give me a hint? $(0;0)$ and $(10;10)$ are obviously solutions. I can't see how to approach the problem, though.
0
votes
1answer
33 views

$f(x)$ is a quadratic function with vertex $(1, −2)$, opens up. [closed]

I got this on a practice quiz. I couldn't figure it out without at least another point on the graph. $f(x)$ is a quadratic function with vertex $(1, −2)$, opens up.
1
vote
1answer
45 views

A quadratic $ax^2+bx+c$ has its roots in the interval $[0,1]$, find the maximum value of $\frac{(a-b)(2a-b)}{a(a-b+c)}$

A friend of mine sent me this problem asking for help, however, I myself, need help, so here am I. I was able to find the minimum of the denominator, which was equal to $b^2/a$ if my steps were ...
2
votes
1answer
35 views

how to find $\frac{a}{b}$ given that $2a^2 + 2007a + 3 = 0$ and $3b^2 + 2007b + 2 = 0$

Given that $$\begin{cases} 2a^2 + 2007a + 3 = 0 \\ 3b^2 + 2007b + 2 = 0 \end{cases} $$ and $ab \ne 1$, how to solve for $\frac{a}{b}$? My try: \begin{align} 2007a = -3 - 2a^2 \\ 2007b = -2 - 3b^2 \\ \...
0
votes
2answers
78 views

How to solve this question of algebra logically? [closed]

solve this. If $ x ^ 2 - x ( 6 - \sqrt 3 ) + 10 - 3 \sqrt 3 = 0 $, then find $$ ( x + 3 ) ^ { 17 } + \frac 1 { ( x + 3 ) ^ { 17 } } \text . $$
0
votes
1answer
17 views

Location of Roots in Quadratic Equation

We learnt about the location of roots of quadratic equation in class today. I have a problem in one case specifically, the one where both the roots are required in the interval $(k_1,k_2)$ where $k_1,...
0
votes
1answer
25 views

Identical roots of a quadratic equation

Let $f(x)=ax^2+bx+c$ and $g(x)=x^2+d$. Now consider $f(x)+\lambda g(x)=0$ It is given that $f(x)+\lambda g(x)=0$ has identical roots and such $\lambda$ values are also identical. 1) if $d>0$ , show ...
0
votes
2answers
57 views

What is the range of the function $f(x) = \frac{7x^{2}-4x+4}{x^{2}+1}$? [closed]

What is the range of $f(x) = \frac{7x^{2}-4x+4}{x^{2}+1}$? I can't find a method to solve this question. I know the answer, but I don't know how to solve it. Any help would be appreciated.
0
votes
1answer
40 views

high school solving quadratics help [closed]

A pilot of a helicopter plans to release a bucket of water on a forest fire. The height $y$ (in feet) of water $t$ seconds after its release is modeled by the movement equation: $$y(t) = -16t^2 -2t+...
0
votes
3answers
43 views

Find the inverse of $x^2+4x+1, [-2,0)$

I am to find the inverse of $x^2+4x+1, [-2,0)$. I am unsure how to 'untangle' the equation in a certain form. I made one step: $$x=y^2+4y+1$$ $$x-1=y^2+4y$$ I realize this should probably be simple. ...
0
votes
0answers
19 views

If the roots of quadratic equation are r and s, what is the value of r-s

Let the roots of quadratic equation $x^2 + px + q$ be equal to $r$ and $s$. Using Vieta's formulas, what is the value of $r - s$ in terms of $p$ and $q$?
1
vote
3answers
47 views

How to isolate $a$ in this equation?

I have the following equation that I would like to isolate the variable $a$ from, but I am a bit stuck. The variable $k$ is known; however, $x$ and $a$ are unknowns. $$ x^2 + \frac{a}{(1-a)} x -\frac{...
0
votes
1answer
47 views

A chord of a parabola

If a chord, which is not a tangent, of the parabola $y^2=16x$ has the equation $2x+y=p$, and midpoint $(h,k)$, then which of the following is (are) possible values of $p, h$ and $k$? $A)\: p=-2, h=2, ...
6
votes
2answers
253 views

Finding $\sum_{n=1}^{\infty} \frac{1}{f(n)}$ where $f$ is a real quadratic function?

Let's consider the following function $f(x)=ax^2+bx+c$, where $a, b$ and $ c $ are all real constants. Is there any way to calculate the value of $$\sum_{n=1}^{\infty} \frac{1}{f(n)}\text{?}$$ I can ...
1
vote
1answer
52 views

For any real number c, the quadratic equation $x^2+x-c^2 = 0$ has two distinct (real) solutions. Is this true or false and explain why.

I am a first year math major taking the introductory proofs course. This is my solution to the question. I would like you to check if my solution is correct or complete. The statement is true. In ...
2
votes
3answers
75 views

finding a relation in $p:p=\frac{1}{3}+\frac{1}{3}\frac{3}{6}+\frac{1}{3}\frac{3}{6}\frac{5}{9}+\cdots$

if $$p=\frac{1}{3}+\frac{1}{3}\frac{3}{6}+\frac{1}{3}\frac{3}{6}\frac{5}{9}+\cdots$$ and $$p^2+ap+c=0.$$ Find $a,c$ also $|c|=2$ My progress:The general term $$T_{m+1}=\frac{(1)(3)\cdots(2m+1)}{(3)(6)...
0
votes
2answers
31 views

How to solve a quadratic equation with Binary coefficients? [closed]

Consider the quadratic equation: $$5x^2 - 50x + 125 = 0$$ It has the roots $x_1 = 5$ and $x_2 = 5$. But now, convert these coefficients into binary: $$101x^2 - 110010x + 1111101 = 0$$ How can I solve ...
0
votes
1answer
43 views

Quadratic equation, cannot solve it while using a technique that doesn't use bhaskara

My teacher taught how to solve squared equations without bhaskara. It's a completely new technique for me. Actually I solve only 2 problems till now. I am having a hard time solving the last one of ...
0
votes
0answers
34 views

Is there an analytical way to solve this system of equations?

I have to solve this system of equations for one class and I have to iterate, which I hate (the class is applied hydraulics). The system is the following: $$h_1-H=a \cdot Q_1^2 $$ $$h_2-H=b \cdot Q_2^...
0
votes
1answer
8 views

Quadratics - Nautical miles and knots question

Ship A is 50 nautical miles west of Ship B. Ship A is heading east at 10 knots and ship B is heading south at 5 knots. Find the minimum distance between the ships, and at what time it occurred What ...
-3
votes
2answers
72 views

Solve the equation $z^3-2z^2+3z-2=0.$ [closed]

Solve the following equation. $$z^3-2z^2+3z-2=0$$ If $a$ is a complex solution of this equation, what does $A$ equal? $$A= \frac{|a|^2}{1-i ^ {43}}$$ It's on my exams and I really need to solve this ...
3
votes
2answers
44 views

Solving a trigonometric equation for purely imaginary numbers [closed]

I'm puzzled with the following exercise: "By constraining $z$ to be purely imaginary, show that the equation $\cos{(z)}= 2$ can be represented as a standard quadratic equation. Solve this ...
0
votes
4answers
97 views

How to prove the following inequality $x+y\ge2$

Let $x$ and $y$ be two real positive integers, such that: $x+y+xy=3$ prove that $x+y\ge2$ I tried some simplifications like this one $x(1+y)=3-y$ and $y(1+x)=3-x$ and using the fact that both of $x$ ...
0
votes
2answers
58 views

Solving a quadratic equation with 3 parameters [closed]

me and my group of students are having trouble solving the following quadratic equation. Any help is appreciated. Thanks in advance.
-1
votes
1answer
16 views

What is the Product of the roots of the first equation?

Both of the following equations have real roots. $$ax^2 +bx+c=0$$ $$(a-b+c)x^2 -2(a-c)x+ (a+b+c)=0$$ If roots of the second equation are α and β show that $\frac{(1-α)(1-β)}{(1+α)(1+β)}$ is the ...
-1
votes
2answers
60 views

How to find the parameter b such that the following sum of quadratic expressions is minimized?

Suppose you have $x_1, ..., x_n$. My task is to find $b \in \Bbb{R}$ such that the sum $\sum_{i=1}^n (x_i - b)^2$ is minimal. Now, I think we can view it as a multivariate function and differentiate.....
0
votes
1answer
21 views

Intermediate Quadratic Equations

If $n$ is a constant and if there exists a unique value of $m$ for which the quadratic equation $x^2 + mx + (m+n) = 0$ has one real solution, then find $n$. Let the roots of the quadratic be $r,s.$ ...
-1
votes
0answers
23 views

Consider whether the equation $T=5Q^2+20Q+110$ is a suitable model. [closed]

Q represents the quantity of newspapers produced in batches of $100000$s. T represents the total cost of production by $\$1000$s. Other follow up questions: What does the model tell you about the ...
0
votes
2answers
95 views

Finding the range of $p$ such that $p = 3 \cos^2 x + 4 \sin x$

Find the range of possible values for $p$ such that $$3 \cos^2 x + 4 \sin x = p$$ I tried: $$\begin{align} p &= \frac{1}{2}(8\sin(x) + 3\cos 2 x + 3) \tag1\\[4pt] &= -\frac32\sin^2x + 4\...
0
votes
3answers
62 views

Show that $x-x^2=\frac{1}{3}$ has no real solutions. [closed]

I have graphed this equation, which shows that there are no real solutions to this equation. How would I go about showing this without relying on a graph?
2
votes
1answer
56 views

Proofs based on convexity for quadratic functions

I have attempted to proof the following, based on the given data: Let $f(x)= x^{T}Ax +2b^{T}x + c$, where $A\in\mathbb R$ symmetric matrix, $b\in\mathbb R$ and $c\in\mathbb R$. Then: i) $x$ is ...
0
votes
1answer
28 views

Absolute values and Quadratic

The difference of the roots of the quadratic equation $x^2 + bx + c = 0$ is $|b - 2c|$. If $c \neq 0$, then find $c$ in terms of $b$. I know Vieta, for sum and product of roots of a quadratic, but am ...
6
votes
2answers
52 views

$f(x),g(x)$,2 quadratic polynomials:$|f(x)|≥|g(x)|∀x ∈ R$. Find the number of distinct roots of equation $h(x)h''(x)+(h'(x))^2=0$ if $h(x)=f(x)g(x)$

Question: If $f(x)$ and $g(x)$ are two distinct quadratic polynomials and $|f(x)|≥|g(x)|$ $∀$ $x ∈ R$. Also $f(x)=0$ has real roots. Find the number of distinct roots of equation $$h(x)h''(x)+(h'(x))^...
-3
votes
2answers
49 views

Modular Arithmetic with Algebra [closed]

If positive integer $x$ satisfies $x^2 - 4x +56 \equiv 14\pmod{17}$, find the minimum value of $x$. I have a solution that uses the quadratic formula, but I'm looking for more elegant ways to arrive ...
0
votes
0answers
60 views

If $ax^2+(b+c)x+d+e=0$ has roots greater than $1$, then show that $ax^4+bx^3+cx^2+dx+e$ has at least one real root [duplicate]

Your are given an equation $$ ax^2+(b+c)x+d+e=0 $$ whose roots are real and greater than $1,$ show that the equation $$ ax^4+bx^3+cx^2+dx+e $$ has at least one real root. Note that $a,b,c,d,e$ are ...
-1
votes
5answers
52 views

Exponential equation question (can't solve)

I came upon this question on a website: Find all the real solutions to $4^x-2^x=56$. I've tried to factor the expression: $2^x(2^x-1)=56$, but I don't know how to proceed. How can I solve this?
1
vote
1answer
62 views

How to systematically find roots of $x^2-x-132 = 0$ with Po-Shen Loh's method?

Po-Shen Loh in his famous video shows how to systematically find quadratic equation's roots. He find the roots for following quadratic equation. $x^2-8x+12 = 0$ Product: 12, Sum: 8 He divides sum ...
2
votes
1answer
213 views

How to stretch this quadratic function

I have a function $$f(x)=\left(\frac{3p}{5d^2}\right)x^2+\frac{2p}{5d}x$$ where $p$ and $d$ are constants. ($x_2$, on the diagram is $d$) Given two points $(x_1,s)$ and $(x_2,p)$ on the function, as ...
2
votes
4answers
58 views

Roots of a quadratic equation.

Assume I have an equation $$a(f(x))^2+bf(x)+c=0.$$ I am required to find the roots of the equation. When I solve this equation I will obtain a value of $x$, say $p$ and that value of $x$ will give me ...
0
votes
1answer
17 views

Minimum value of RSS: Why does the the coefficient of $d^2$ being positive tell us that this value is a minimum?

I am currently studying the textbook Statistical Inference by Casella and Berger. Chapter 11.3.1 Least Squares: A Mathematical Solution says the following: For any line $y = c + dx$, the residual sum ...
3
votes
2answers
64 views

Given two real numbers $x,y$ so that $x^{2}+y^{2}+xy+4=4y+3x$. Prove that $3\left(x^{3}-y^{3}\right)+20x^{2}+2xy+5y^{2}+39x\leq 100$.

Given two real numbers $x, y$ so that $x^{2}+ y^{2}+ xy+ 4= 4y+ 3x$. Prove that $$3\left ( x^{3}- y^{3} \right )+ 20x^{2}+ 2xy+ 5y^{2}+ 39x\leq 100$$ I used derivative and Wolfram|Alpha but only the ...
-1
votes
1answer
27 views

Figuring out coefficients of composition of a first degree polynomial into a quadratic

given $g(x)=x^2 + x-2$ and $ g \circ f = 2 [2x^2-5x +2]$, find $f(x)$ ( $f(x)$ is form $ax+b$) I found the inverse of g as two functions, $$ y =( x + \frac12)^2 - \frac94 $$ $$ \pm (\sqrt{y + \...
3
votes
0answers
47 views

An optimization problem related to parabolas, yields a hard to solve derivative

Hello, I have came up with what I think is a unique optimization problem. We are given the positive real parameters $k,t$. $t$ is the height of the rectangle, and $k$ is half of its width (see the ...
0
votes
1answer
22 views

Region of the coefficients of a quadratic equation that cause the roots of it to be in the unit disk

From Simon Haykin's Adaptive Filter Theory: consider the characteristic equation is $1+a_{1}z^{-1}+a_{2}z^{-2}=0$, then for the roots to be inside the unit circle (or in the unit disk), the ...
1
vote
4answers
76 views

Given that $a,b,c$ satisfy the equation $x^3-2007 x +2002=0$, then find $\frac{a-1}{a+1}+\frac{b-1}{b+1} +\frac{c-1}{c+1}$

Given that $a,b,c$ satisfy the equation $x^3-2007 x +2002=0$, then find $\frac{a-1}{a+1}+\frac{b-1}{b+1} +\frac{c-1}{c+1}$ The concept of transformation of roots can be applied here. So replace $$x \...

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