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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2
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3answers
81 views

How can you find $m$ in $mx^2+(m-3)x+1=0 $ so that there is only one solution

How can you find $m$ in $$mx^2+(m-3)x+1=0 $$ so that there is only one solution. I tried to solve it by quadratic equation but I end up with two solutions. So I want it know that is there a way so ...
3
votes
1answer
289 views

Can this nonlinear simultaneous equation be solved?

Problem: $\{A,B,E,R,S\}\in R^{n \times n}$ are square matrices, $\{\mathbf{x},\mathbf{y}\}\in R^{n}$ are vectors. Particularly, $\{ A,B \}$ are symmetric matrices, and $E$ is an identical matrix. We ...
0
votes
2answers
69 views

Solve the following number theory problem with 2 variables [closed]

Let there be $$a,b∈ \Bbb Z$$ Demonstrate that there exist no solutions for the following equation $$a^2-3b^2=-1$$
1
vote
1answer
98 views

How do the roots of “$x^2 + bx + c$” change as $b$ is kept constant and $c$ is changed? [closed]

Consider the function $x^2 + bx + c$ How do the (real or complex) roots of the equation change if $b$ is held constant and $c$ is changed? I.e. Which patterns are evident? What would it look like if ...
0
votes
0answers
15 views

Show that $p(x)=rq(x)$ for some rational number $r$.

Let $p(x)$ and $q(x)$ be two quadratic polynomials with integer coefficients. Suppose they have a non rational root in common. Show that $p(x)=rq(x)$ for some rational number $r$.
0
votes
2answers
34 views

For odd integers $a,b,c$ line $ax+by+c=0$ cannot intersect parabola $y=x^2$ in a rational point

For odd integers $a,b,c$ line $ax+by+c=0$ cannot intersect parabola $y=x^2$ in a rational point(where both abscissa and ordinate are rational numbers.) We need to solve the equation of the line ...
40
votes
12answers
5k views

Why do I get one extra wrong solution?

I'm trying to solve this equation: $$2-x=-\sqrt{x}$$ Multiply by (-1): $$\sqrt{x}=x-2$$ power of 2: $$x=\left(x-2\right)^2$$ then: $$x^2-5x+4=0$$ and that means: $$x=1, x=4$$ But $1$ is not a ...
2
votes
2answers
54 views

How do I factorise this difficult quadratic without a calculator?

On one of the UKMT maths challenge past papers(Team challenge) it asks you this question: Factorise $120x^2 + 97x - 84$ That is the whole question. I used a calculator and found that you factorise ...
1
vote
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
vote
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 ...
3
votes
1answer
42 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 ...
0
votes
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 ...
6
votes
2answers
124 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 ...
0
votes
3answers
85 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
votes
1answer
39 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 ...
0
votes
1answer
32 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.
0
votes
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. ...
0
votes
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 ...
1
vote
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, ...
1
vote
1answer
56 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 ...
0
votes
1answer
64 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 = ...
1
vote
1answer
41 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 ...
0
votes
1answer
42 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 ...
4
votes
0answers
66 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)) ...
0
votes
2answers
36 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 ...
0
votes
2answers
40 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 ...
0
votes
2answers
30 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
votes
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 ...
1
vote
2answers
29 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, ...
1
vote
2answers
26 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
votes
0answers
73 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 ...
1
vote
2answers
58 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 ...
1
vote
2answers
61 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 ...
1
vote
2answers
82 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. ...
1
vote
3answers
38 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 ...
1
vote
2answers
33 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
vote
3answers
91 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: $$ ...
0
votes
6answers
53 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
votes
1answer
82 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 ...
1
vote
3answers
72 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
votes
1answer
82 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, ...
0
votes
1answer
40 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: ...
2
votes
2answers
27 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 ...
0
votes
2answers
35 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 ...
5
votes
3answers
95 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 ...
6
votes
3answers
76 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
votes
4answers
78 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
votes
1answer
175 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
votes
1answer
35 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{ ...
1
vote
9answers
298 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 ...