Questions on 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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9
votes
3answers
531 views

Quadratic Formula, nature of roots with Trigonometric Functions

The original problem: If $0\le a,b\le 3$ and the equation $$x^2+4+3\cos(ax+b)=2x$$ has at least one real solution, then find the value of $a+b$ $$$$ At first, on rearranging, I got the ...
1
vote
0answers
39 views

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 ...
1
vote
3answers
41 views

Calculating range of values of $k$ s.t. the graph $y=4x^2-kx+25$ does not cut or touch the $x$ axis

Calculate the range of values of $k$ so that the graph $y=4x^2-kx+25$ does not cut or touch the $x$ axis. I just don't know what to set delta to as I can't work out if the graph would be a tangent ...
1
vote
1answer
54 views

Quadratic Equation; Roots' Magnitude Less than 1

What are the conditions on $a$ and $b$ so that the roots (real or complex) of the equation have magnitude $< 1$. $$λ^2 − (a − b + 1)λ + a = 0$$ On a separate note, if you could explain (NOT ...
4
votes
2answers
45 views

Find the value of $\sqrt[4]{\alpha}-\sqrt[4]{\beta}$,where $\sqrt[4]{.}$ denotes the principal value.

If $\alpha$ and $\beta$ are the roots of the equation $x^2-34x+1=0$,find the value of $\sqrt[4]{\alpha}-\sqrt[4]{\beta}$,where $\sqrt[4]{.}$ denotes the principal value. I found out the $\alpha$ ...
1
vote
5answers
39 views

quadratic equation expressed as a range of values

Solve $6 - x - x^2 < 0$ A. $-3 < x < 2$ B. $x < -3, x > 2$ C. $-2 < x < 3$ D. $x < -2, x >3$ So this is a very easy quadratic to solve: $x^2 + x - 6 &...
1
vote
1answer
64 views

Resolving the tedious cubic

The equation given to me is $$4x^4 + 16x^3 - 17x^2 - 102x -45 = 0$$ I'm asked to find it's resolvent cubic which is not so difficult to find. But the problem is that the question further asks to find ...
4
votes
2answers
51 views

Maximize the sum $a+b$

If the equation $x^4-4x^3+4x^2+ax+b=0$(where $a,b$ are reals) has $2$ distinct positive roots $\alpha, \beta$ such that $\alpha+\beta =2\alpha \beta$, then find the maximum value of $a+b$. I have no ...
3
votes
2answers
37 views

If $a\in R$ and the equation $-3(x-\lfloor x \rfloor)^2+2(x-\lfloor x \rfloor)+a^2=0$ has no integral solution,then all possible values of $a$

If $a\in R$ and the equation $-3(x-\lfloor x \rfloor)^2+2(x-\lfloor x \rfloor)+a^2=0$ has no integral solution,then all possible values of $a$ lie in the interval $(A)(-1,0)\cup(0,1)$ $(B)(1,2)$ $(C)...
-1
votes
2answers
23 views

Express $s = -5t^2 + 40t$ in the form of $a(t-b)^2 + c$, where $a$, $b$ and $c$ are the constants.

$s= -5t^2+ 40t$. Express $s$ in the form of $a(t-b)^2 + c$, where $a$, $b$ and $c$ are the constants. $s = -5t(t-8)$. I have factorized it.
0
votes
5answers
62 views

Solving $ \sqrt{5x+1}+\sqrt{x-1}=2$

How to solve: $$ \sqrt{5x+1}+\sqrt{x-1}=2$$ I can tell that 1 is a solution but I am not sure how to solve this algebraically, do i start by squaring both sides?
4
votes
2answers
123 views

How can we prove that a quadratic equation has at most 2 roots?

A quad equation can be factored into two factors containing $x $, but how can we prove that there no other sets of different factors yielding OTHER VALUES OF $X $?
1
vote
1answer
130 views

Shortest distance between two circles

What is the shortest distance, in units, between the circles $(x - 9)^2 + (y - 5)^2 = 6.25$ and $(x + 6)^2 + (y + 3)^2 = 49$? Express your answer as a decimal to the nearest tenth. So I know that ...
1
vote
1answer
48 views

Given a set of $(x,y)$ coordinate pairs how can I come up with the equation for a curve?

Say for example I have a set of coordinate pairs as follows: x y 0 100 20 90 100 0 I would like to generate an equation for this curve In the ...
0
votes
2answers
46 views

The quadratic spline

I'd like to fit the data in table as blow x f(x) 3.0 2.5 4.5 1.0 7.0 2.5 9.0 0.5 when $x=5$, I want to find value of $f(x)$ by using ...
2
votes
1answer
45 views

Least Squares Fitting Quadratic Equation to a set of points

My math skills from my college days are a bit rusty, so if my terminology is wrong, I apologize. I will try to be as clear as I can. I have a set of 50 points on a plane that roughly follow the ...
0
votes
1answer
13 views

How to solve for a & b in a parabolic equation?

Not a mathematician at all here, actually a programmer and this came up... I need some help on something that is pretty basic I'm guessing. Given the following formula how would I go about solving ...
0
votes
5answers
84 views

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 $$3x^2+22x+24$$
0
votes
1answer
17 views

Finding parameter for quadratic equation

Given $x^2 - 3ax + a^2 = 0$ and $$\frac{x_1^4-x_2^4}{\sqrt{5}x_1x_2} + x_1 + x_2 -20x_1x_2 - 4 = 0$$ Find $a$. The answer is $1$ ($a = 1$) I tried to present $x_1^4 - x_2^4$ as $(x_1+x_2)(...
2
votes
3answers
47 views

Why is the graph of a quadratic function a parabola?

I'm sorry for the stupid question, but it seems that extensive googling didn't yield an answer. I've learned about parabolas, and how the parabola is the curve that is equidistant from a point (Focus)...
1
vote
0answers
68 views

If $a+b+c = 0$ then the quadratic equation $3ax^{2}+2bx +c=0$ has atleast one root in _________?

If $a+b+c = 0$ then the quadratic equation $3ax^{2}+2bx +c=0$ has atleast one root in _________? Rolle's theorem states that if $f(a) = f(b)$ then there exists a $p \in [a,b]$ such that : $f'(p) = \...
6
votes
2answers
61 views

Quadratic equation with one root in $[0,1]$ and other root in $[1,\infty]$

Find the values of $a$ for which $x^2-ax+2=0$ has one root in $[0,1]$ and other root in $[1,\infty]$. The twoo rots are $$\frac{a\pm\sqrt{a^2-8}}{2}$$ The smaller root should be less than $1$. So $$...
1
vote
4answers
80 views

How would one solve the following equation?

This equation is giving me a hard time. $$e^x(x^2+2x+1)=2$$ Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph ...
2
votes
2answers
40 views

Quadratic expression problem involving sides of a triangle

If $a,b,c$ be the sides of a triangle where $a\neq b\neq c$ and $\lambda \in R$, if roots of the equation $x^{2} + 2\lgroup a+b+c\rgroup x + 3\lambda\lgroup ab+bc+ca\rgroup =0$ are real, then what is ...
1
vote
2answers
50 views

Pairs of Quadratic equations

If each pair of the three equations $$x^2 + P_1x + q_1 = 0$$ $$x^2 + P_2x + q_2 = 0$$ $$x^2 + P_3x + q_3 = 0$$ have a common root, then prove that $$P_1^2 + P_2^2 + P_3^2 + 4(q_1 + q_2 + q_3) = 2 ( ...
0
votes
1answer
47 views

How to graph a Quadratic equation.

I have an equation that goes: $0.0001x^2 - 0.22x + 197$. I'm not asking for the answer, but instead, how can I graph it without dealing with these insanely tough numbers.
0
votes
2answers
20 views

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!
2
votes
2answers
67 views

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in G.P, find $x$.

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in Geometric Progression, find $x$; $x \neq 0$. Here, $\{x\}=x-\lfloor x\rfloor$ Some properties are pretty evident: $$0\leq \{x\} < 1 \tag{1}$$ $$\...
0
votes
2answers
23 views

Problem with graphical representation of quadratic equation.

I have an equation $x^2 -x =0$ I know, it has two solutions, $x=0,1$ If I plot it using a math software, it shows a straight line passing through $1$, perpendicular to x-axis. (I have plotted, $x^2-x=...
0
votes
1answer
16 views

Proof quadratic congruent equation has no solutions in $\mathbb{N}$

In computer science, quadratic probing is used in hash tables. We choose a $c_1$ and $c_2$ in the hash formula $h(k,i) = (h'(k) + c_1 i + c_2 i^2) \mod{m}$ where $h'(k) = k \mod{m}$ and $m$ is the ...
5
votes
0answers
88 views

Probability that the roots of a quadratic equation are real

Roots of the quadratic equation $x^2+5x+3=0$ are $4\sin^2\alpha+a$ and $4\cos^2\alpha+a$. Another quadratic equation is $x^2+px+q=0$ where $p,q\in\mathbb{N}$ and $p,q\in[1,10]$. Find the probability ...
0
votes
1answer
27 views

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 $...
0
votes
1answer
29 views

How to solve nonlinear system generated from $(x_i - x_j)^2 \approx f_{ij}$

Looking for advice/help with solving a nonlinear system generated from the equation: $(x_i - x_j)^2 \approx f_{ij}$ where $\textbf(X) = (x_1,x_2,x_3,...,x_n)$, $f_{ij}$ are known, solve for $\textbf(...
0
votes
1answer
17 views

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 ...
0
votes
0answers
29 views

Roots of $f(x)=0$ and $g(x)=0$ are imaginary

Let $f(x)=ax^2+bx+c$ and $g(x)=Ax^2+bx+D$, where $a$ and $A$ are non-zero and $a,b,c,A,D\in R$ Roots of $f(x)=0$ and $g(x)=0$ are imaginary, then which of the following may be correct: (A) $f(x)+g(x)=...
0
votes
1answer
32 views

Quadratic inequalities 1

The question is : $2x^2-3x-1 \leq 0$ I've got the answers but I keep getting them wrong. If someone could write out the steps it would be greatly appreciated.
1
vote
0answers
16 views

Given Cartesian points A and B, how to move A by n units towards B? [duplicate]

I have two points, for which I know Cartesian coordinates. I want to move one towards the other over specific distance. That's about the same as saying I want the distance between the points equal ...
0
votes
2answers
42 views

Trigonometric equation solving / 4th degree polynomial

Okay, so I was solving a free body diagram problem, no need to send it, but I found a very huge problem in doing so, in any way I tried solving , I either got to an equation $\sin(x-0.11)=4\sin(2x)$ ...
1
vote
2answers
48 views

Quadratic equation

Here's the question below - $x^2 - 6x + (p^2 - 6 )^2$ is a perfect square , write down the possible values of p . My thoughts : I thought of this expansion method - $(A-B)^2 = A^2 - 2 AB + B^2$ ...
1
vote
2answers
52 views

Sketching the graph from function

I am having some issues with sketching the following graph. Any assistance is appreciated. The function $g(x) = -x^2 + 6x + 4$ , $K \leq x \leq 6$ where $K$ is a constant. Also, $g^{-1}(x)= -(y+3)^2 ...
0
votes
0answers
32 views

A simple proof involving a quadratic equation.

Am I doing this proof right? I am asked to prove $$\exists z \in \mathbb{R} \forall x \in \mathbb{R}^{+}[\exists y \in \mathbb{R}(y-x=y/x)\iff x \ne z]$$ Proof: Letting $z = 0$ (I figured that $x \...
2
votes
2answers
106 views

How to prove that the roots of this equation are integers?

Let there be an equation $a^2 + 4ab + b^2 - 121 = 0$ where I want to prove that a,b are integers. Then I want to find whether there are integer values of $b$ for which $a$ is also an integer. Let us ...
3
votes
1answer
46 views

Total number of values of $a$ such that the equation $x^2+ax+a+1=0$ has integral roots

Find the total number of values of $a$ such that the equation $x^2+ax+a+1=0$ has integral roots The equation can also be written as $$\left(x+\frac{a}{2}\right)^2=\frac{a^2-4a-4}{4}$$ So $a$ is a ...
0
votes
1answer
37 views

When do I solve a quadratic expression by either factorising, completing the square or use the quadratic formula?

What are the rules and conditions as to which method I should use. I know how to use them, just not when I should.
1
vote
2answers
59 views

Existence of real roots for vector quadratic equations

I have a vector quadratic equation of the form $\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x} + \boldsymbol{x}^{T} \boldsymbol{b} + c = 0$ where $\boldsymbol{A}$ is symmetric and for my particular ...
-3
votes
4answers
65 views

Simplify in the form: $x^n + \frac{1}{x^n}$ [closed]

Simplify: $$\left(x^2 + \sqrt2 + \frac{1}{x^2}\right)\left(x^2 - \sqrt2 + \frac{1}{x^2}\right)$$ in the form $x^n + \frac{1}{x^n}$
2
votes
3answers
33 views

Indices and Bases: Solve “x”

Solve the equation $2^x - 3^{x-1}=-(x+2)^2$ How I got this question? I created this question so I know the answer. The answer is 5. But I have no idea how to solve it. Take note that I cannot do ...
0
votes
2answers
30 views

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 ...
0
votes
2answers
39 views

Inverse of $y=2x^2-12x+13$

I'm having a problem finding the inverse of $y=2x^2-12x+13$. At the end I get to the following: $$x=3 \pm \frac{\sqrt{40+8y}}{4}$$ As far as I know the answer is suppose to be $x= 3 \pm \frac{\sqrt{...
1
vote
2answers
59 views

possible real solutions of the equations

What are the possible real solutions of the equations $$1000=v_1^2+4v_2^2,100=v_1+4v_2$$ Its a physics question but I thought its not necessary to post here . Thank you.