Questions on quadratic equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-h)^2+k$ or $y=a(bx+c)(cx+d)$.

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0answers
11 views

Calculating $\arg\min_x (1-\Phi(x;\mu_1,\sigma_1^2)+\Phi(x;\mu_2,\sigma_2^2))$

I would like to find $x$ satisfying the following expression: $$\arg \min_x R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2)$$ where $$R(x,\mu_1,\mu_2,\sigma^2_1,\sigma^2_2) ...
0
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0answers
15 views

Ellipse equation. What does it need to be in order for $b > a$?

We have the quadratic equation: $$ax^2 + bx + cy^2 + dy + e$$ $a$ and $c$ are both negative or both positive. How can I, by looking at that only, determine whether $b$ (the length of the semi-minor ...
2
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4answers
31 views

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take?

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take? This is what I have done: Let $y = 3^x$ $$9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$$ $$\implies9y^2 + (t^2 - 4t - 2)y + ...
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1answer
24 views

Help on Quadratic Equations [on hold]

If $\sin15$ and $\cos 15$ are the roots of a quadratic equation $x^2+ax+b=0$, then find the value of $a^4- b^2$. Please, need help, show working, thanks.
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2answers
33 views

How factor with square root

I have the following equation that I'm trying to factor, but I'm stuck at the end. $$\frac{zx^{-4}\sqrt{x}(yz^4)^3}{z^7xy}$$ $$\frac{\frac{1}{x^4}\sqrt{x}(yz^4)^3}{z^6xy}$$ ...
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3answers
20 views

Need help with basic factoring equation

I'm just trying to brush up on my factoring of quadratic equations. $$\frac1{x+3} + \frac1{x^2 + 5x +6}$$ $$\frac1{x+3} + \frac1{(x+2)(x+3)}$$ $$\frac{(x+2)(x+3) + (x+3)}{(x+2)(x+2)(x+3)}$$ Then ...
3
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1answer
53 views

Factors of integers of the form $k^2-k+1$

Factorisation of arbitrary integers is of course a computationally hard problem. But what if the integers I'm interested in factorising are all of the form $k^2-k+1$ ? Is there some way to compute ...
2
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0answers
47 views

Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real root…

Problem : Consider the quadratic equation $ax^2-bx+c=0, a,b,c \in N. $ If the given equation has two distinct real roots belonging to the interval $(1,2) $ then the minimum possible values of a is ...
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2answers
45 views

Roots of the quadratic equation, $\alpha$ and $\beta$

Given that $\alpha$ and $\beta$ are the roots of the equation $4x^2 -6x +1$. Find $\alpha^3-\beta^3$.
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2answers
24 views

Quadratic equation form?

Suppose we know that the sum of two positive numbers is $2k$ and their product is $m$ then which of the following will be its quadratic equation and why? 1) $x^2$+ $(2k)x$+ $m$= $0$ 2) $x^2$- ...
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4answers
60 views

When do you use the quadratic formula?

I am revising for a mathematics exam and am looking over simultaneous equations. I was curious as to when I use the quadratic formula and when I don't? I realize there are multiple ways to solve a ...
1
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1answer
35 views

how to solve these two quadratic equations

Can someone help me find the solution for these two quadratic equations ? $ 2(z^2) \ - \ 3.023bz \ + \ 0.115(b^2) \ + \ 2.0814b \ + \ 0.142z \ - \ 0.5856 \ = \ 0 $ $ 6.0828(z^2) \ + \ 2.0414bz \ + \ ...
0
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1answer
15 views

What is the minimum possible $ non $ integral value of a

Let a A subscript(m) (m=1,2,3,....p) be the possible integral values of a for which the graphs of $ f(x)=ax^2+2bx+b $ and $g(x)=5x^2-3bx-a$ meets at some point for all real values of b. 1) What is ...
3
votes
1answer
137 views

Finding the shortest path length on a curved surface(hyperboloid)

I wish to find the minimum path length between two points $P_1(\sqrt2,0,-1)$ and $P_2(0,\sqrt2,1)$ on a hyperbolic surface $S =\{(x,y,z)\in R^3\ |\ x^2+y^2-z^2=1\}$ I faintly recall studying ...
1
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1answer
19 views

A question on multinomial theorem using binomial theorem

$(3x^2+2x+c)^{12}=\sum A_r x^r$ and $\frac{A_{19}}{A_5}=\frac 1 {2^7}$ Find $c$. I really have no idea what to do with this. This was on a test. I have studied only binomial theorem. So, please ...
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1answer
25 views

Determining quadratic function of this word problem

I have this word problem in my homework: ...
1
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1answer
56 views

A Cubic Equation

$2x^3+ax^2+bx+4=0$, $(a,b \in R^+)$ has three real roots. Then : A. $a\geqslant 4.2^{\frac 1 3}$ B. $a\geqslant 1.2^{\frac 1 3}$ C. $a\geqslant 6.2^{\frac 1 3}$ D. $a\geqslant 2.2^{\frac 1 3}$ ...
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1answer
15 views

Terminology of “linear”, “quadratic”, etc. for multi-input functions

It is my understanding that, according to typical math terminology: The function $f(x, y) = x + y$ is "linear". Specifically, it's linear in both $x$ and $y$, but this is understood implicitly. ...
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2answers
57 views

number of integral values for which $x^2+19x+92$ is a perfect square.

number of integral values of x for which $x^2+19x+92$ is a perfect square=? I have no idea how to do this. Please help.
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0answers
20 views

Variable quadratic functions

I have some doubts in proving the following:- C is the curve $$y = \frac 1 {k+1} [2x^2 + (k + 7)x + 4]$$, where k is a real number not equal to -1. Show that C always passes through two fixed points ...
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2answers
112 views

How would I solve the quadratic $x^2+3x-70=0$?

How would I solve the following quadratic equation $$x^2+3x-70=0 $$ This is my attempt below $$(x-7x) (x+10x)=0 $$ $$ x-7x=0 \implies -6x=0 \implies x=6$$ $$x+10x=0 \implies 11x=0 \implies ...
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1answer
31 views

How to interpret coefficients in polynomial regression?

I am working on my thesis (study) about poverty incidence rate and its socio-economic factors using second-order polynomial regression without interaction. The final model in my study is ...
0
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1answer
18 views

Verification of solutions to some polynomial prob/

$\boxed{\text{Problem 1}}$ Find the other solution of the equation $(1+\sqrt3)x^2-(5-\sqrt3)x+6-6\sqrt3=0$ given that $2$ is a solution Ma solution: $x_1\cdot x_2=\tfrac ca$ therefore letting ...
2
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2answers
30 views

prove for p(x) which is a quadratic polynomial

$p(x)$ is a quadratic polynomial . Prove that any given number for $a$ with one exception , we can find a number $b$ such that $p(a)=p(b)$ and $a$ is not equal to $b$.
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2answers
90 views

Solving for $a$ and $b$

How would you solve the equality: $$a\,b\,\left(b+a\right)=1$$ in terms of a and/or b? Would you subtract 1 from both sides and work from there? Or would you simply expand and work from there?
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2answers
35 views

How to use $t(29/\sqrt{2})<0$ where $t(x)=x^2-41x+420$ to prove that $41/29<\sqrt{2}<42/29$??

So I was investigating different ways to approximate $\sqrt{2}$. Here's my latest: $$Let:t(x)=x^2-41x+420$$ then the roots of $t(x)$ are $20$ and $21$. I showed that then $t(x)=(x-20)(x-21)$ and ...
4
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2answers
70 views

Integral values of an expression

Let $b=\sqrt{a^2+5a+8}-\sqrt{a^2-3a+4}$ Find number of integral values of b. My $long$ way using Calculus : Find domain of function : $R$ Note that function is continuous Prove the function is ...
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2answers
27 views

What does each term in the trinomial represent and their relation to each other? [duplicate]

I am trying to get a grasp or concept understanding what how trinomials answer questions other than answering questions in an algebra class. I. Looking for the practical application. What does the ...
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1answer
30 views

With Trinomial's, can someone explain the purpose of the first, second and third term. In layman terms.

I know that the first term is a quadratic and I suppose that lets us know we are dealing with identifying a curve, and the third term is our constant. I just can't quite put it all together as how ...
2
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2answers
30 views

Find the range of values of $k$ for which the following equation has real roots.

Find the range of values of $k$ for which the following equation $x^2+(1-k)x=k$ has real roots. I tried it, for real roots, $b^2-4ac \geqslant 0$ $x^2+(1-k)x-k=0$ $(1-k)^2-4*1*(-k)\geqslant0$ ...
20
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10answers
3k views

Why is every equation always equal to zero? [closed]

A linear equation is $$ ax + b = 0 ; \,\, \,\, a\neq 0 $$ A quadratic equation is $$ax^2 + bx + c = 0 ; \,\, a\neq 0 $$ And so on... Why are all equal to zero? Why have mathematicians defined it ...
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2answers
39 views

Solving quadratics using $y=a(x-p)^2+q$ ?

The vertex is $(-4, 2)$ and the y-int is $(-3,-1)$. How would I solve this, or find out what "$a$" is so I can write the equation and graph it?
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3answers
70 views

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$.

Find the range of values of $x$ for which $1-x<(x-1)(5-x)<3$. First of all, I solved $1-x<(x-1)(5-x)<3$ which gives me $(x-1)(x-6)<0$ and $(x-4)(x-2)<0$. How to find the range, ...
1
vote
1answer
48 views

Found an example for solving via quadratic formula in a book where I am wondering if this is correct

As a refresher, I was skimming through a free Calculus online textbook "MOOCULUS massive open online calculus" (https://mooculus.osu.edu/handouts) and stumbled upon the following example solving a ...
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2answers
34 views

Find the range of values of $x$ for the inequality $x^2-4x-1>0$ [closed]

Find the range of values of $x$ for the inequality given. $x^2-4x-1>0$
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5answers
29 views

Linear systems from positions a quadratic passes through?

I don't understand the question. $$y = ax^2 + bx + c$$ passes through the points $(1,-4),(-1,0),(2,3)$. Write down a linear system (unknowns $a,b,c$) of three equations relating the unknowns to each ...
0
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5answers
102 views

Show that $3x^2-4x+2$ is always greater than $0$.

How do I show that the function $3x^2-4x+2$ is always greater than $0$?
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1answer
117 views

Find the maximum or minimum value of the quadratic function by completing the square.

Find the maximum or minimum function of the quadratic function by completing the squares. State the value of $x$ at which the function is maximum or minimum. $y=3x^2+7x+9$ I already posted similar ...
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2answers
175 views

Find the maximum or minimum value of the quadratic function.

Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of $x$ at which the function is maximum or minimum. $y=2x^2-4x+7$ $x^2$ has a coefficient ...
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4answers
50 views

Convert the L.H.S to the form of the R.H.S by completing the square.

Given that $x^2-3x+5=(x-p)^2+q$ for all values of $x$, calculate the value of $p$ and of $q$. A book example tells me to firstly convert the L.H.S to the form of the R.H.S by completing the square. ...
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3answers
34 views

Find the value of $p$ and $q$ of the quadratic equation.

The quadratic equation $x^2+px+q=0$ has roots $-2$ and $6$. Find the value of $p$ and $q$. Do I have to make two equations? Something like this? When $x=-2$, (real and distinct roots) ...
0
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1answer
46 views

Inequality challenge

I was studying inequations when I encountered this problem here. How can I find a region of values for m where this inequation is true? $$-3<\frac{x^2+mx-2}{x^2-x+1}>2$$ Thanks
1
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1answer
29 views

Find the possible values of $p$ for which the equation has coincident roots.

Find the possible values of $p$ for which the equation $(2p+3)x^2+(4p-14)x+16p+1=0$ has coincident roots. Coincident roots means 'equal roots'. For equal roots, we should use: $b^2-4ac=0$ ...
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2answers
22 views

all the values of c for the equation: x^2-5x+c>2

We need to find all the values of c for this equation: x^2-5x+c>2 This question was on my exam I didn't know how to get started on the question. Could some one help me out.
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2answers
54 views

Find the possible values of $k$, if the equation has equal roots.

The equation $x^2+5k=kx+x+19$ has equal roots. Find the possible values of $k$. Um having problem in rearranging the equation; $x^2+5k-kx-x-19=0$ $x^2+k(5-x)-x-19=0$ What is the next step?
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3answers
45 views

Solving a quadratic using vietas theorem I keep going in circles.

I am trying to solve the quadratic equation $x^2-48x+432=0$ with out directly factoring OR using the quadratic equation. I am going with vieta. So $$r+s=48$$ and $$rs=432$$ I've already solved it by ...
0
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0answers
33 views

Solving quartic equations

I am solving quartic equations by finding the intersections of two quadratics. The given quartic function is of the form $x^4 + px^2 + qx + r = 0$. I know one of the quadratic functions to be $y=x^2$. ...
0
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0answers
40 views

Given the matrix, find c such that det(D)=0 has a repeated solution

Given the matrix $D= \begin{vmatrix} 1 & x & x^2\\ 2 & c & 4\\ 3 & 2 & 1 \end{vmatrix} $ Find c such that $\det(D)=0$ has a repeated solution for $x\in R$. I got up to ...
0
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1answer
48 views

How to turn algebra into quadratic equation? [closed]

I have to find the maximum value of $x^2+(5+x)^2$ using a parabola. How do I turn the $x^2+(5+x)^2$ into a quadratic equation?
1
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1answer
77 views

Solve …

This is what I did Can anyone tell me what's wrong me or the question?