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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3answers
67 views

Solve this complex number: $z^2+(1+i)z+i=0$

I need to solve: $$z^2+(1+i)z+i=0$$ first of all I used $$z = (a+ib)$$ and I get: $$(a+ib)^2+(1+i)(a+ib)+i=0$$ $$a^2-b^2+2aib+a+ib+ia-b+i=0$$ then I have ordered, on the left the number without ...
1
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2answers
44 views

Why are these 2 algebraic expressions equivalent?

I just solved a long problem for my physics w/calculus homework that required a simplification using a quadratic formula. The "textbook" (flipItPhysics) came up with a different simplification than ...
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4answers
39 views

Determine if there is a solution in inequality [closed]

Does this inequality have a solution? $$x^2 < 5x - 6$$ I try to solve but after I check, the solution is wrong.
3
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1answer
34 views

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$

Let $a,b,c,d$ be distinct integers such that the equation $(x-a)(x-b)(x-c)(x-d)-9=0$ has an integer root $r$,then find the value of $a+b+c+d-4r.$ As $r$ is the integer root of the equation ...
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1answer
28 views

Calculate the quadratic residues in Z∗17.

Hello I am wondering if any one can help me I am trying to figure out how these below answers where came to too. Calculate the quadratic residues in Z∗17. Solution: This can be done by direct ...
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0answers
117 views

Question about an equation using linear algebra

Firstly, in $n \in \bf{N}$, we consider the square matrices as $\bf{\{H_{i},A_{i},P,Q \}} \in \bf{R}^{\it{n \times n}}$ and the vectors $\bf{\{ a,x,1 \}} \in \bf{R}^{\it{n}}$. All parameters are known ...
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2answers
119 views

Looking for a function $g(x)$ such that $g(2x+2) = g(x) + 2x+2$

So recently I got bored in maths class (I'm in tenth grade) and made up a little equation that looked something like this: $$g(f(x)) = g(x) + f(x) $$ My original goal was to find different $g(x)$ to ...
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3answers
84 views

No. of parabolas possible for the given equation.

Given, $f(x)=-x^2+qx+r$. $(q,r) \epsilon R$. $q,r$ are variables. A quadratic equation $f(x)=0$ has a maximum value $m$ ($m$ is a constant) and a root $x=a$. Does $f(x)$ correspond to a unique ...
3
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1answer
38 views

Easy word problem but am I working it too hard?

A friend of mine gave me this problem: A man who walks at a constant speed goes to his barn 30 miles away with a 2 mph wind pushing against him. After arriving at the barn he remembers he ...
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1answer
24 views

Quadratic number pattern equation

May I know how do I form a quadratic number pattern equation? I cant seem to form one on my own. 1500, 1519,1536, 1551,1564.
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0answers
37 views

Did I fully explain this optimization and quadratics problem?

I'm not really sure how to explain the last part; how does solving for $x$ by replacing $y$ show that $x^2+y^2$ is greater than or equal to $9?$ Like, I get why, but I don't know how to express it. ...
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0answers
30 views

How to invert a transformation

I've come across a recursive equation involving vectors. You basically have one starting point $P = (x, y)$ and you transform it to another point $P'=(x', y')$ with the following equations $$ x' = x ...
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1answer
39 views

Quadratic equations amount

Quadratic equation looks like that: $$ax^2+bx+c=0$$ where $a\ne 0$. We can say something about roots when We compute a discriminant $$\Delta=b^2-4ac$$ When $\Delta>0$ then We have two real roots, ...
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1answer
41 views

What will create a box with largest volume?given image

Jamal wants to make a box with no top out of a 24 inch square piece of cardboard. She plans to cut smaller squares of equal size from the corners of the cardboard and fold up the resulting sides. To ...
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1answer
63 views

Solve in $\mathbb{C}$ the equation $z^2-(1+m)(1+i)z+i(m^2+1)=0$

Hi I tried to solve the equation $$ z^2 - (1+m)(1+i)z + i(m^2+1)=0 $$ but I don't know if my answer is wrong or right. My first $\Delta$ was $-2(im^2+i-2im)$, the second one $0$. So $$ z_1 = ...
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1answer
34 views

How to find a quadratic equation given three points, two on the x-axis?

Find the quadratic equation for a parabola that passes through $$(1,0) (5,0) (0,10)$$ To do this I turned it into $$ x = 1 $$ $$ x = 5 $$ and then into $$(x-1)(x-5)$$ after you multiply everything ...
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1answer
40 views

How to solve the following quadratic word problem given a quadratic equation?

The height of a ball(h), in feet, after s seconds is modeled by the equation $$h=-16t^2+40t-6$$ How many seconds does it take for the ball t reach its maximum height? First thing i did was turn the ...
3
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0answers
56 views

Quadratic equation too hard

I am trying to solve a quadratic equation very hard. Is there any other way to solve this without quadratic formula? $$ x^2(-BE(F+C)^2(G+C)(A+C))))+x(C(F+C))\left [ EBA(D-H)-(G+C)(A+C)(B(D-H)+D(F+C)) ...
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2answers
28 views

How to solve a word problem when given width and height of the following?

The width of a room is 4 feet shorter than its length, and its height is 3 feet less than its length. The area of four walls is larger than the sum of the areas of the floor and ceiling by 134 square ...
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2answers
39 views

Which equation has roots -2c, 2c, and 2?

This is a multiple choice question $$-4c^2 -2c=0$$ $$-4c^2+2c=0$$ $$x^3 - 2x^2-4x+8=0$$ $$x^3 - 2x^2-4c^2x +8c^2=0$$ I know roots mean solutions, so do I plug in the given roots and see if they ...
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2answers
29 views

Any solution of the inequality,$\frac{\log_3(x^2-3x+7)}{\log_3(3x+2)}<1$ is also a solution of the inequality,$x^2+(5-2a)x-10a\leq 0$

Find out the values of $a$ for which any solution of the inequality,$\frac{\log_3(x^2-3x+7)}{\log_3(3x+2)}<1$ is also a solution of the inequality,$x^2+(5-2a)x-10a\leq 0$ I first found out the ...
0
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1answer
25 views

how to solve the following quadratic word problem *updated*?

The total cost of carpeting a rectangular room is given the expression $$6x^2 + 18x$$ Which situation best describes the expression? The length of the room is 6+2x feet, its width is 2x feet, and ...
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2answers
24 views

Find all the values of the parameter $c$ for which the inequality has atleast one solution.

Find all the values of the parameter $c$ for which the inequality has atleast one solution. $$1+\log_2(2x^2+2x+\frac{7}{2})\geq\log_2(cx^2+c)$$ First i checked the domain of the inequality, ...
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2answers
24 views

If the inequality $\log_a(x^2-x-2)>\log_a(-x^2+2x+3)$ is known to be satisfied for $x=\frac{9}{4}$ in the interval $(x_1,x_2)$

If the inequality $\log_a(x^2-x-2)>\log_a(-x^2+2x+3)$ is known to be satisfied for $x=\frac{9}{4}$ in the interval $(x_1,x_2)$,then find the product $x_1x_2$. Here $a$ is not specified .I know ...
3
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0answers
61 views

Let $\alpha$ and $\beta$ be the roots of a quadratic equation $4x^2-(5p+1)x+5=0$.If $\beta=1+\alpha,$then find the integral value of $p.$

Let $\alpha$ and $\beta$ be the roots of a quadratic equation $4x^2-(5p+1)x+5=0$.If $\beta=1+\alpha,$then find the integral value of $p.$ Sum of roots$=\alpha+\beta=\frac{5p+1}{4}$ Given ...
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2answers
54 views

Find the range of $m$ for which atleast one of the following equations $ax^2+bx+cm=0,bx^2+cx+am=0,cx^2+ax+bm=0$ have real roots.

Let $a,b,c$ and $m\in R^+$.Find the range of $m$ for which atleast one of the following equations $ax^2+bx+cm=0,bx^2+cx+am=0,cx^2+ax+bm=0$ have real roots. Either one or two or all of the three ...
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2answers
58 views

How to solve the following quadratic word problem?

The total cost of carpeting a rectangular room is given the expression $$6x^2 + 18x$$ This is the multiple choice type question so the given options were set up like this. The length of the room ...
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2answers
71 views

Find the unique pair of real numbers $(x,y)$ that satisfy $P(x)Q(y)=28$

Let $P(x)=4x^2+6x+4$ and $Q(y)=4y^2-12y+25$. Find the unique pair of real numbers $(x,y)$ that satisfy $P(x)Q(y)=28$ I can solve this question graphically. ...
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3answers
35 views

If $m,M$ are the minimum and maximum value of $\alpha^2+\beta^2$,then find $m+M.$

Let $\alpha,\beta$ be real roots of the quadratic equation $x^2-kx+k^2+k-5=0$.If $m,M$ are the minimum and maximum value of $\alpha^2+\beta^2$,then find $m+M.$ I calculated ...
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2answers
32 views

If $2$ is subtracted from each root,the results are reciprocals of the original roots.Find the value of $b^2+c^2+bc.$

The equation $x^2+bx+c=0$ has distinct roots .If $2$ is subtracted from each root,the results are reciprocals of the original roots.Find the value of $b^2+c^2+bc.$ Let $\alpha$ and $\beta$ are the ...
1
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3answers
88 views

Solving an equation involving the sum of square roots of a quadratic

So I am building a computer program. In the program I need to build a function that takes this arguments {a, b, c, d, e, f, s, u}, and returns back the value of x in this equation: $$ ...
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6answers
49 views

Finding constants of a given curve

Find constants $a$, $b$ and $c$ such that the curve $y= ax^2 + bx +c$ passes through the point $(0,3)$ and has a relative extremum at $(1,2)$? I tried substituting the values of the given coordinates ...
3
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1answer
77 views

Is This a Proof of Multiplication On The Parabola?

I am a high school student who is beginning to look at proofs and I was wondering if this could be considered a proof for a property of multiplication of points on a parabola. I've seen this result ...
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3answers
65 views

Find the range of values for k such that ${kx^2 + 3x + 9k = 0}$ has real roots

I am asked the question: Find the range of values for ${k}$ such that ${kx^2 + 3x + 9k = 0}$ has real roots. So from my understanding, there are distinct roots if ${b^2 - 4ac\ge 0}$ My first step ...
3
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1answer
77 views

If $\frac{x^2+ax+3}{x^2+x+a}$ takes all real values, prove $4a^3+39<0$

If $\frac{x^2+ax+3}{x^2+x+a}$ takes all real values for possible real values of $x$, then prove that $4a^3+39<0$. Here is how I approached it. Let $$\frac{x^2+ax+3}{x^2+x+a}=y$$ Then, ...
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1answer
38 views

Solving a system of quadratic equations which evaluates to a 4th grade equation

I have to solve the following system of equations: $x^2 + 4y + 2 = 22$ $2y^2 + x + 6 = 40$ I tried to solve for one variable and then substitute it into the other equation, but a problem appears: ...
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2answers
26 views

Find quadratic equation given two points and y-intersection

Let the domain be $x \in [0,h]$. We have three points, $(0,1)$, $(h/2,0)$ and $(h,0)$. How do I find the quadratic equation? My attempt: I know that the roots are located at $x=h/2$ and $x=h$. Thus ...
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2answers
28 views

Using the absolute value when taking the square root in an inequality

I have a question about why the unknown becomes absolute when taking the square root in an inequality. For example: Find the value(s) of $k$ for which the equation $2x^2-kx+3=0$ will have two ...
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3answers
91 views

If $x-y = 5y^2 - 4x^2$, prove that $x-y$ is perfect square

Firstly, merry christmas! I've got stuck at a problem. If x, y are nonzero natural numbers with $x>y$ such that $$x-y = 5y^2 - 4x^2,$$ prove that $x - y$ is perfect square. What I've ...
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3answers
67 views

Finding the values of $q$ for which the quadratic equation $qx^2-4qx+5-q=0$ will have no real roots.

Find the values of $q$ for which the quadratic equation $qx^2-4qx+5-q=0$ will have no real roots. So I've gotten as far as using the discriminant to find the values of $q$, but I'm stuck on the last ...
2
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4answers
72 views

Why does this method to solve a quadratic equation for $x$ omit $x=0$?

Here is a simple quadratic equation: $$9x^2 - 36x = 0$$ We proceed as following: \begin{align*} 9x^2 & = 36x\\ 9x & = 36\\ x & = 4 \end{align*} So, we get $x=4$. But, here's another ...
7
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1answer
162 views

Geometric derivation of the quadratic equation

The quadratic equation can be thought of as specifying distances in the Euclidean plane. It tells us that the $x$-intercepts of a function occur at a distance of $\frac{\sqrt{b^2-4ac}}{2a}$ from the ...
3
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1answer
34 views

Why are the factors of some solutions to a Pell equation also a solution?

I came across this observation while trying to answer this post using the Pell equation $x^2-2y^2=1$. Define, $$P(m) = \frac{ (3+2\sqrt{2})^m+(3-2\sqrt{2})^m}{2}$$ $$Q(m) = \frac{ ...
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9answers
295 views

Why is $x=2 \implies (x-2)(x-3)=0$ false?

Let $P(x)$ be the equation $x=2$ and $Q(x)$ be the equation $(x-2)(x-3)=0$ By definition of implication I see that $P(x)$ implies $Q(x)$... As I see it, any premise that is false can give any ...
2
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2answers
68 views

Derive quadratic formula [duplicate]

I cannot understand how quadratic formula to solve for $x$ was derived. On this website, it explains the steps Following I understand but I cannot understand how they got $b/2a$ and why they ...
12
votes
1answer
75 views

Aside from the obvious stuff, do the partial functions that solve the quadratic equation have any interesting properties?

Let us define partial functions $$f_+,f_- : \mathbb{R} \leftarrow \mathbb{R} \times \mathbb{R} \times \mathbb{R}$$ so as to return the zeros of the quadratic $ax^2+bx+c$ whenever they exist, such ...
3
votes
2answers
56 views

Complex number in quadratic equation

Find $a,b$ given that a root of $x= 1+2i$ and the equation $ x^2+(a+bi)x+2i-1=0$ I tried finding it by $\Delta$, which I got $\Delta=a^2+2abi-b^2-8i+4$ I tried substituting the root into the equation ...
0
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2answers
28 views

Given Function, find domain and description of graph $y = f(x)$

I am studying for Graduate Record Exam. The following question is difficult. Given the domain and description of $f(x) = 5 - (x + 20)^2$, including its shape, and the $x$ and $y$-intercepts To find ...
0
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1answer
32 views

Counting the solutions of a quadratic equation

I have read that a non-singular conic will contain $p+1$ points on the finite field $\mathbb{F}_{p}$, but there is a exercise on Silverman's Rational Points on Elliptic Curves, p.142, 4.8 that tells ...
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2answers
29 views

Bound on Coefficients

For real $a,b,c$ the following holds $|ax^2+bx+c|\le 1 ; \forall x\in [0,1]$.Show that $|a|+|b|+|c|\le 17$. Cant show that the equality holds.I always get the lesser bounds.