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

2
votes
1answer
23 views

Range of rational expression having algebraic terms

Find the range of $\displaystyle f(x) = \frac{x^2+x-1}{x^2-x+2}$ subjected to $-1 \leq x\leq 1$. Plan \begin{align} y = \frac{x^2+x-1}{x^2-x+2}&\implies yx^2-yx+2y=x^2+x-1\\&\implies(y-1)x^2-...
0
votes
4answers
24 views

Quadratic equation and two points.

I need to solve a quadratic equation (actually I need to explain it to my kid), but I get stuck in the middle and would be grateful, for any pointers into the right direction. $y=ax^2+bx-1$ with two ...
1
vote
2answers
33 views

To find Range of roots of quadratic equation

I have been given the following quadratic equation and is asked to find the range of its roots $\alpha$ and $\beta$, where $\alpha>\beta$ $$(k+1)x^2 - (20k+14)x + 91k +40 =0,$$ where $k>0$ . ...
2
votes
1answer
72 views

Name for this method of factoring quadratic and are there any textbooks that describe it?

I remember learning this method of factoring quadratics in middle school or high school, but looking for a name or more information on it leads me to dead ends. Given: $ax^2+bx+c=0$ $d*e=a*c$ $d+...
1
vote
2answers
37 views

The values of $x_1 \cdot x_2$ from $x^{2}-2mx+2m^{2}-2m=0$

I have the following equation: $x^{2}-2mx+2m^{2}-2m=0$ with the real roots $x_1,x_2$ I need to find the values of the product of $x_1\cdot x_2$.The right answer is $[-\frac{1}{2},4]$ My try: To have ...
2
votes
3answers
43 views

An Extra Solution

I hope you're well, I was finding the points of intersection of $$ x^2 + (y-1)^2 = 1 \quad \text{and} \quad y = 1-x^2 $$ If I rearrange the formula of the circle to $$ (y-1)^2 = 1-x^2 $$ then ...
-2
votes
1answer
22 views

Let (x+a) be the HCF of $x^2+px+q$ and $x^2+mx+n$. Show that $a=(q-n)/(p-m)$ [closed]

Let $(x+a)$ be the HCF of $x^2+px+q$ and $x^2+mx+n$. Show that $a=(q-n)/(p-m)$.
1
vote
2answers
33 views

How to prove that $|f(\frac pq)| \geq \frac {1}{q^2}$ for $f(x) = ax^2 + bx + c$ with real roots and coefficients as odd positive integers

The question says: If $f(x) = ax^2 + bx + c$ have real roots and it's coefficients are of positive integers, then (A) $f(x) = 0$ always have real roots (B) $\left|f\left(\frac pq\right)\...
6
votes
4answers
636 views

Is it possible to determine $P(21)$, if $P(x)$ is a 2nd degree polynomial, $P(11)=151$, and for all $x\in\Bbb R$, $x^2-2x+2\le P(x)\le2x^2-4x+3$?

I was given the following: Given that $P(x)$ is a second degree polynomial, and that $P(11)=151$, and that $$\forall x \in \mathbb{R}, \,\, x^2-2x+2 \le P(x) \le 2x^2-4x+3$$ determine $P(21)$. ...
0
votes
4answers
35 views

Basic Function Question

Can someone please help me , I’ve no idea how to do this: Give an example of a quadratic function $f$ that satisfies $f(x) ≤ 0 ⇔ x ∈ (−∞,−5) ∪ (\frac{7}{2},∞)$.
0
votes
2answers
51 views

Algebra - Factoring Quadratic Equations

I'm in algebra and this problem was under the lesson Factoring to Solve Quadratic Equations. The problem is the following: The product of two consecutive numbers is 14 less than 10 times the ...
0
votes
2answers
51 views

For what value of $p$ does $(p^2-16)x^2-(p^2-4p)x+(p^2-5p+4)=0$ have more than $2$ solutions in the variable $x$?

The answer key suggest that solution is (d) $p=4$ but I think that if $p=4$ then it do not remain quadratic equation since for being quadratic equation square coefficient should be non zero but for $p=...
1
vote
2answers
37 views

How to solve this quadratic congruent equation by inspection

I found a systematic way (c.f. How to solve this quadratic congruence equation) to solve all congruent equations of the form of $ax^2+bx+c=0\pmod{p}$, or to determine that they have no solution. But ...
1
vote
2answers
77 views

Comparing two variable expressions quickly

Let us say $S_1=2xy^2+3xy$, and $S_2=3y^2+7x+7y+8$. Then, can we say that $S_1\ge S_2$ if $y\le x-2$ and $x,y\in\mathbb{N}$? I think yes, but the usual quadratic function method is taking too much ...
1
vote
2answers
34 views

How do you solve $\frac{x-1}{\sqrt{x}+2}=\frac{5}{2}$?

I solved it using the quadratic formula and it went like: \begin{gather} \frac{x-1}{\sqrt{x}+2}=\frac{5}{2} \\ 2(x-1)=5(\sqrt{x}+2) \\ 2x-2=5\sqrt{x}+10 \\ 2x-12=5\sqrt{x} \\ 4x^2+144-48x=25x \tag*{(...
-1
votes
1answer
42 views

Pell's Equation and Continued Fractions [closed]

For each of the following equations, determine whether there are no solutions, finitely many solutions, or infinitely many solutions with $x, y$ justify your answers. $$x^2-5y^2=3 \\\ x^2+7y^2=...
0
votes
2answers
23 views

Why is the $2a$ term in the quadratic formula $|2a|$?

I am reading through Algebra by Gelfand/Shen. There was a construction of the quadratic formula as follows: $ax^2 + bx + c = 0$. Dividing by $a$ gives us $x^2 + \frac{b}{a}x + \frac{c}{a} = 0$ and ...
0
votes
1answer
12 views

Solve for $\tau$ where variable inside square root.

How do you solve for $\tau$? $$t = \tau + \frac{\sqrt{A^2+v^2\tau^2}}{c} $$ it might be easy but I just can not see how. Thanks.
0
votes
2answers
36 views

Quadratic equation with natural number coefficients

Let $a,b,c $ be Natural Numbers, such that roots of the equation $ax^2-bx+c=0$ are distinct and both lie in the interval (0,1) (1,2) (2,3) (Brackets signify open interval, roots are $IN BETWEEN $ ...
0
votes
1answer
29 views

Find $y$ if $x^{x+y}=y^n$ and $y^{x+y}=x^{2n}y^n$, where $x,y,n>0$

If $x,y>0$ satisfying the system of equations $x^{x+y}=y^n$ and $y^{x+y}=x^{2n}y^n$, where $n>0$ then prove that $y=\dfrac{1+4n-\sqrt{1+8n}}{2}$ $$ (xy)^{x+y}=(xy)^{2n}\implies x+y=2n\\ x^{2n}=...
1
vote
1answer
24 views

Not able to find roots of linear 2nd order homogeneous dif. equation

I am trying to find the roots of the differential equation $y''+7y'+6y=0$. I assume the following: $$ y=e^{rx} \\ y'=r*e^{rx} \\ y''=r^2*e^{rx}. $$ Then I substitute that into the dif. equation: $...
0
votes
2answers
15 views

Quadratic equation with weighted average coefficients

I have tried using the quadratic formula as well as factoring method to solve the following quadratic equation but failed to get the correct answer. The equation is: $$ \theta x^2-x+(1-\theta)=0.$...
2
votes
2answers
72 views

Solve $2x^2-5x+2=$ $\frac{5-\sqrt{9+8x}}{4}$

Solve $2x^2-5x+2$= $\frac{5-\sqrt{9+8x}}{4}$ I simply do square both sides solve it and I get two value of x one is 2 and other is $\frac{3-√5}{2}$ but this approach it take more time so is there any ...
3
votes
7answers
117 views

How would you explain this method of solving quadratic equations?

I stumbled across this interesting geometrical method of solving quadratic equations. Can someone explain why are intersection points roots of equation? Why does circle have anything to do with ...
0
votes
2answers
50 views

Quadratic equation including Arithmetic Progression

For $a, b, c$ are real. Let $\frac{a+b}{1-ab}, b, \frac{b+c}{1-bc}$ be in arithmetic progression . If $\alpha, \beta$ are roots of equation $2acx^2+2abcx+(a+c) =0$ then find the value of $(1+\alpha)(1+...
0
votes
4answers
512 views

How was the quadratic formula created? [duplicate]

I have tested the quadratic formula and I have found that it works, yet I am curious as to how it was created. Can anybody please tell me one of the ways that it was created?
0
votes
3answers
38 views

Quadratic square values

Find the value(s) of positive integer $n$ such that $n² + 19n + 48$ is a perfect square. I factorised it to $(n+3)(n+16)$, but that gives negative integer answers $-3$ and $-16$. What do I do?
-2
votes
2answers
29 views

finding A and B while only product and sum are given [duplicate]

Question: If A + B = 54 and `AB = 629`, find A and B I am not sure how to approach this problem since the question itself does not give much clue.
0
votes
1answer
32 views

Prove $f(x)=0$ has one root in between roots of $g(x)=0$

$f(x)=(a-b)x^2+(b-c)x+(c-a)$ $g(x)=(b-a)x^2+(a+c-2b)x+(b-c)$ given that $a<b $ and $2a^2+b^2+ac<3ab+bc$ Find common root of $f(x)=0 $ and $ g(x)=0$ . Prove $f(x)=0$ has one root in between ...
5
votes
2answers
92 views

Real roots of the equation $\log_{(5x+4)}(2x+3)^3-\log_{(2x+3)}(10x^2+23x+12)=1$

Find the set of real roots of the equation$$\log_{(5x+4)}(2x+3)^3-\log_{(2x+3)}(10x^2+23x+12)=1$$ My Attempt $$ 2x+3>0, 5x+4>0, 2x+3,5x+4\neq1\implies x>-4/5\;\&\;x\neq -1\;\&\;x\neq ...
0
votes
4answers
42 views

Solve for all real $x$: $(3x^2-8x+5)/(4x^2-3x+7) > 0$ [closed]

I know how to solve quadratic inequalities but I can't figure out how to solve $$\frac{3x^2-8x+5}{4x^2-3x+7} > 0$$ for all real $x$. Help would be appreciated
4
votes
6answers
366 views

Solving $3x^2 - 4x -2 = 0$ by completing the square

I can't understand the solution from the textbook (Stroud & Booth's "Engineering Mathematics" on a problem that involves solving a quadratic equation by completing the square. The equation is ...
0
votes
1answer
43 views

If $ a, b, c$ are real numbers such that $a^2 + b^2 + c^2 = 1$, then show $ab+bc+ca> \frac{-1}{2}$

If $ a, b, c$ are real numbers such that $a^2 + b^2 + c^2 = 1$, then show that $ab+bc+ca\ge \frac{-1}{2}$ If figured out that if I put $(a+b+c)^2 = 0$ then I will get the above answer, but $(a+b+c)^...
1
vote
1answer
14 views

If $\frac{x^2-bx}{ax-c} = \frac{k-1}{k+1}$ has roots, whose magnitude is equal but signs are opposite.

If $\frac{x^2-bx}{ax-c} = \frac{k-1}{k+1}$ has roots, whose magnitude is equal but signs are opposite. Answer is $\frac{a-b}{a+b}$ I used cross multiplication and since the roots are opposite in ...
0
votes
3answers
61 views

Find the minimum value of $x$ in the given that $\frac{\sqrt{2x^2-1} + \sqrt{x^2-1}}{\sqrt2x^2}=1$

How to simplify the given equation and find the minimum value of $x$ ? $$\frac{\sqrt{2x^2-1} + \sqrt{x^2-1}}{\sqrt2x^2}=1$$ I do square both sides but I doesn't make any sense
1
vote
2answers
53 views

Minimum value of $\frac{2-\cos(x)}{\sin(x)}$ without differentiation

I have to find the minimum value of the expression $$\frac{2 - \cos x}{ \sin x}$$ Also $x$ lies between $0$ to $\pi$. One way is to find the minima using differentiation. But it is not taught in my ...
0
votes
1answer
20 views

Are these coefficients swapped in this example, or am I wrong?

I believe I'm staring at an error here. Attached is a screenshot. Am I right in understanding that the author swapped the coefficients around in solving q using the quadratic formula? I can see why ...
1
vote
1answer
32 views

Alpha-beta quadratic equation

Question: The equation $$3x^2-6x-4=0$$ has roots α and β. Find the value of 1/α + 1/β. I'd just like confirmation on my answer, as I've already found the answer but am not confident in it. since αβ=...
0
votes
1answer
38 views

Turning the two solutions of a quadratic equation in a general solution

So while I was working on a project for uni, I came across this quadratic equation: $$t^2 + 2\tau t - 1 = 0$$ The solutions for this particular equation are: $$t_{1, 2} = \tau \pm \sqrt{\tau^2+1}$$ I ...
0
votes
1answer
32 views

How to solve the quadratic equation with 2 unknown parameters for P as a function of w?

I tried to solve the following equations: $$-w^2 + 11w -11/2 = 10(w-P)-(w-P)^2$$ First, I got rid of the brackets and ended up with the following equation: $$w-11/2 = 2wP-10P-P^2$$ And now I am ...
2
votes
1answer
41 views

Range of rational function

How to find the Range of function $$f(x)= \frac{x^2+2x-3}{x^2 - 3x +2}$$ I made a quadratic equation in terms of y which came to be: $$ y(x^2 - 3x +2)= x^2+2x-3 $$ $$\implies x^2(y-1)-x(3y+2)+2y+3=...
-1
votes
0answers
12 views

Active set method algorithm

I'm not so sure about the active set method for solving an optimization method. So the problem is $\min J = \frac{1}{2} (x, Ax) - (f, x) $, that becomes $Ax = f $ since A is SPD. The constraint is $u \...
0
votes
2answers
29 views

How do I work out an unknown probability? (Sorry if I'm not specific enough details in main body)

There are two blue beads and x red beads in a box. The probability that two random beads taken at random from the box are both red is 15/22, how do I work out x?
1
vote
1answer
69 views

Decomposing bivariate quadratic form into sum of two squares

I would like to decompose $$ax_1^2 + bx_2^2 + 2cx_1x_2$$ into two expressions, each involving only one variable. I'm trying to use a transform like $x_1 = x_+ + x_-$ and $x_2 = x_+ - x_-$ to ...
-1
votes
3answers
27 views

Analytical solution for equation with power $n \in [1, 2]$

Does there exist analytical solution to the equation $ax^p + bx + c = 0$ where $p \in [1, 2]$? Please provide references if the answer is affirmative.
2
votes
1answer
62 views

Minimising quadratic objective subject to logarithmic inequality constraints

I need to minimise the following function: $$ (2a_1 + 2a_2 - 1)^2 + (2a_1 + 2a_3 - 1)^2 $$ subject to: $$ \sum a_i \log_2 a_i \geq -1 $$ where all the $i \in \{1,2,3,4\}$ and $a_i \in [0,1]$ and $\sum ...
1
vote
4answers
76 views

System of quadratic equations with three variables (generic form)

Try solve a system of equation like this one. \begin{cases} (O_x -A_x)^2+(O_y-A_y)^2+(O_z-Az)^2=x^2 \\ (O_x -B_x)^2+(O_y-B_y)^2+(O_z-Bz)^2=y^2 \\ (O_x -C_x)^2+(O_y-C_y)^2+(O_z-Cz)^2=z^2 \end{cases} ...
2
votes
5answers
55 views

Solving quadratic equation $x^2-(k+1)x+k+1=0$ using quadratic formula

Disclaimer I am new here so please mind how i have executed this question. Feel free to comment on how to improve as i have read on “how to ask a good question”. Problem I am having trouble ...
0
votes
3answers
52 views

given $x^2 + y^2 = 2x$. I want $(x-1)^2 + y^2 = 1$

given $x^2 + y^2 = 2x$. I want $(x-1)^2 + y^2 = 1$ Is this completing the square? my attempt: $$x^2 + y^2 - 2x = 0$$ need $1$ $$x^2 + y^2 - 2x + 1 = 1$$ I forgot how to could someone explain ...
0
votes
3answers
43 views

Check if true : Atleast one of the integers, a, b, c must be even

Suppose a, b, c are integers such that the equation $ ax^2 + bx + c =. 0 $ has a rational root. Check if true : Atleast one of the integers, a, b, c must be even. I know for rational roots $ b^2 - ...