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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5
votes
2answers
458 views

How to solve this equation? Can I treat as a quadratic equation?

$$\ln(x+3)+\ln(x-4)=0$$ How to solve this equation? First removing the 'ln' from the equation and after making a quadratic equation and then solve the quadratic equation?
0
votes
1answer
37 views

If the roots of $x^2+x-1$ are $\alpha$ and $\beta$, find an $eq^{n}$ whose roots are $\alpha^{19}$ and $\beta^{7}$

If the roots of $x^2+x-1$ are $\alpha$ and $\beta$, find an $eq^{n}$ whose roots are $\alpha^{19}$ and $\beta^{7}$ My Procedure The roots are $$\frac{-b+\sqrt{b^{2}-4ac}}{2a}$$ and ...
-2
votes
1answer
40 views

List the elements of the set $\{X \in \mathbb Z \mid 4X^2 +11X = 0\}$ [closed]

I don't get this maths equation Can anybody explain it ? Thanks List the elements of the following set: $A=\{X \in \mathbb Z \mid 4X^2 +11X = 0\}$.
2
votes
1answer
20 views

Evaluating cubic roots of a quadratic

If $\alpha$ and $\beta$ are the roots of the quadratic equation $2x^2 + 4x -5 = 0$, evaluate $\alpha^3 + \beta^3$.. I know that $$\alpha + \beta = \frac{-b}{a}$$ and $$\alpha \beta = ...
1
vote
2answers
28 views

Factor the Quadratic

-16t^2+32t+20=0. How are you supposed to find -5 and positive 1 to put in the parenthesis? -4(2t-5)(2t+1)?
-1
votes
3answers
48 views

Inequalities - x^2 - 1/2 x - 5 < 0 ; why is x > 2 1/2?

Question : $$\text{ find the set of values of }x \text{ for which } $$ $$10 + x - 2x^2 < 0$$ Answer : $$x < -2$$ $$x > 2\frac{1}{2}$$ EDIT - thanks for the responses. To try and ...
2
votes
1answer
94 views

Quadratic Irrationality of the Periodic points of the Gauss map

If $G:[0,1] \rightarrow [0,1]$ is the Gauss map which is defined as $$G(x) = \left\{\frac{1}{x}\right\} = \frac{1}{x} - \left\lfloor\frac{1}{x}\right\rfloor,$$ show that if $x$ is periodic of order ...
0
votes
1answer
48 views

Issue on proving quadratic formula

I have come across a stage of the proof: $$ \left(x+\frac b{2a}\right)^2=\frac{b^2-4ac}{4a^2}$$ How does $\left(x+\frac b{2a}\right)^2$ not equal $\pm x\pm \frac b{2a}$ when taking the square root?
0
votes
1answer
21 views

Inequalities and equations - creating sets from quadratic equations.

My question is just making sure that my working is correct and that I understand properly (self teaching, can get confused...) So question : Find the set of values for which $$x^2 -4x-12 < 0$$ ...
0
votes
3answers
57 views

Quadratic equation $9x^2-37=6x$ using the quadratic formula

Quadratic equation using the quadratic formula $9x^2-37=6x$ So $9x^2-6x-37=0$ $A= 9$ $b=-6$ $c=37$ $\dfrac{-(-6) \pm \sqrt{ (-6)^2- 4(9)(37)}}{2(9)}$, $\dfrac{6 \pm \sqrt{36-1332}}{18}$, $\dfrac{6 ...
1
vote
2answers
52 views

Quadratic equation $4x^2+4x=7$ using quadratic formula

Solve using quadratic formula. $4x^2+4x=7$ So $4x^2+4x-7=0$ $A=4$ $b=4$ $c=-7$ $$x=\frac{-4\pm\sqrt{(4)^2-4(4)(-7)}}{2(4)}=\frac{-4\pm\sqrt{16+112}}{8}=\frac{-4\pm\sqrt{128}}{8}$$ What's next?
1
vote
3answers
66 views

Factorize $6x^2 -5x -14 = 0$

I'm throwing a bit of a blank on the best way to factor this : $$6x^2 -5x -14 = 0$$ I know that I could multiply $6$ by $14$ and then find a pair of factors that add to $-5$ (b), but this feels a ...
0
votes
1answer
56 views

intersection of 4 circles

Hi I'm doing some programming challenges for fun and I am trying to work out the maths required to solve this problem. It has been 10 years since I did any maths in anger like this so i'm a bit ...
0
votes
2answers
19 views

Help with demonstration of formula for the axis of a parabola

At school we are studying the parabola and our teacher said that the formula for the axis of a parabola is $x=-\frac{b}{2a}$ without giving us the demonstration; so I tried to come up with a nice ...
0
votes
0answers
14 views

Changing Variables Method for Solving a Quadratic Equation

I am reading a book that contains different ways of deriving the quadratic equation. One of the methods that it discusses is "Changing the Variables." It contains an exercise that I don't understand: ...
0
votes
1answer
34 views

Quadratic $y = -4.9x^2 + 25x$

Here is my questions, please help. In the game of foot, a team can score by kicking the ball over a bar and between two uprights. For a kick in a particular game, the height of the ball above the ...
1
vote
2answers
29 views

Prove the given condition from given two quadratic equation

Question: If the quadratic equations $x^2+bx+c=0$ and $bx^2+cx+1=0$ have a common root then prove that either $b + c + 1 = 0$ or $b^2 + c^2 + 1 =bc + b + c$ Till yet, I had figured the common ...
1
vote
1answer
25 views

Checking some work on finding roots

OK, I have the following response function: $$H(\omega) = \frac{1-\omega^2 LC}{1+\omega^2 LC - i \omega RC}$$ I want to find where it becomes $\frac{1}{\sqrt{2}}$. This should be simple enough. ...
0
votes
1answer
36 views

Simplification of another nasty expression

I have the following condition $$ 2 \frac{x^2}{y^2} \left(1 - \frac{1}{y^2} \right)+ \frac{1}{y^2} \leq 1$$ Can anyone help me simplify it to the best possible relationship between $x$ and $y$?
0
votes
1answer
19 views

Find the range of values $k$ can take given that, for real $x$, $f(x) = \frac{x^2+3k}{x+k}$

I'm trying to find the range of values $k$ can take given that, for real $x$, $f(x) = \frac{x^2+3k}{x+k}$ can take any real value. These are the steps I've taken so far: $$ xy + ky - x^2 - 3x = 0 $$ ...
0
votes
2answers
20 views

Find the range of $k$ in $f(x) = \frac{x^2-k}{x-2}$

I have the following question: For real $x$, $f(x) = \frac{x^2-k}{x-2}$ can take any real value. Find the range of values $k$ can take. Here is how I commenced: $$ y(x-2) = x^2-k \\ -x^2 + xy - ...
-1
votes
1answer
78 views

If the quadratic equation $x^2 + 2kx + 2(k + 4) = 0$ has distinct real roots, then $k^2 – 2k – 8 > 0$ [closed]

The quadratic equation $x^2 + 2kx + 2(k + 4) = 0$ has distinct real roots. Show that $k^2 – 2k – 8 > 0$. I'm not sure what you're meant to do here- it's a 2 mark question.
3
votes
1answer
63 views

Why doesn't this method of solution work?

Solve $$\sqrt{2x^2 - 7x + 1} - \sqrt{2x^2 - 9x + 4} = 1 \tag1$$ I tried to do the following: $$(2x^2 - 7x + 1) - (2x^2 - 9x + 4) = 2x-3\tag2$$ Dividing $(2)$ by $(1)$ yields $$\sqrt{2x^2 ...
-2
votes
2answers
40 views

Completing the square Quadratics

Solve this quadratic equation by completing the square: $2x^2+x-4=0$ Can I have the method aswell please.
0
votes
2answers
23 views

expanding and simplifying an expression

so the question is $(2x - 2)^2 + (3 - 2x)^2$. My working out: $$ 2x (3 - 2x)^2 + 2 (3 - 2x)^2 = 6x^2 - 4x + 6 - 4x = 2x^2 - 4x + 6. $$ I was a bit confused as their was another way to work out this ...
0
votes
2answers
35 views

Solving quadratic equations in the field $F_5$

Let $y = x^2 + 2x + 2 = 0$. Solve the equation in the field $F_5$. So I used the common $b^2 - 4ac$ formula and got that $x$ is either $-1/2$ or $-3/2$ but I'm not sure if this is in the field...
0
votes
1answer
32 views

factor polynomial as $(1-x\phi)(1+x\phi)$ instead of $(x-\phi)(x+\phi)$

I'm reading Generatingfunctionology, which is turning into a bit of an algebra review for me, and I am stumped by a step on page 9. I see the quadratic $1-x-x^2$ and I just pull the minus out ...
2
votes
0answers
46 views

$A(x+p)²-B(x-p)²=y$, historical/math reference

I'm trying to build a reminder of all that I found about the quadratic function over the years. Here I came across this form of quadratic equation that I did not know: A(x+p)²-B(x-p)²=y I have no ...
0
votes
2answers
40 views

Basis for the Space of Quadratic Polynomials $P^{(2)}$ — Homework Help

Prove that $1+t^2$, $t+t^2$, $1+2t+t^2$ is a basis for the space of quadratic polynomials $P^{(2)}$. I have worked it out to the point where I have the following: $(1+t^2)(1, 0, 1)^T ...
2
votes
0answers
29 views

can someone break this quad formula down for me?

Can someone explain how this person yield the stuff on the right side using quad formula?
0
votes
4answers
35 views

After completing the square.

After completing the square, what are the solutions to the quadratic equation below? $$x^2 + 2x = 25$$ Honstely I think it's B. But I'm not sure.
1
vote
2answers
35 views

Solve the equation, find the x

I need to find the x in thid equation. How is it done? a = x+1/x I've tried turning it into x² = 1-x times a, but it's not a system.. so.. any ideas?
0
votes
1answer
36 views

Parabola - How far from the thrower does the ball strike the ground?

The height of a ball thrown in the air is given by $h(x) = \frac {–1}{12} x^2 + 6x+ 3$, where x is the horizontal distance in feet from the point at which the ball is thrown. c. How far from the ...
0
votes
1answer
32 views

Find the dimensions of a square piece of cardboard given data of it folded into a square (cubic inches, etc.)

A box with a square base and no top is to be made from a square piece of cardboard by cutting 6 in. squares out of each corner and folding up the sides. The box needs to hold 1000 in3 . How big a ...
0
votes
2answers
68 views
0
votes
0answers
42 views

Quadratic Formula: Box, Word Problem.

An open-topped box is being made from a piece of cardboard measuring 12 in. by 30 in. The sides of the box are formed when four congruent squares are cut from the corners, as shown in the diagram. The ...
1
vote
1answer
21 views

Quadratic Formula: Word Problem.

A person decides to build a horse corral using a barn for one side. Has has 30m of fencing materials and wants the corral to have an area of 100m^2. What are the dimensions of the corral? Let width ...
1
vote
3answers
45 views

Under what conditions will one solution of $ax^2+bx+c = 0$ be the reciprocal of the other?

Under what conditions will one solution of $ax^2+bx+c = 0$ be the reciprocal of the other?
0
votes
2answers
62 views

Quadratic equations word problem.

A uniform walkway is built around a rectangular flower bed that is 20m by 40m. There is enough material to make a walkway that has a total area of 700 m^2. What is the width of the walkway? I need ...
0
votes
0answers
10 views

parabolic function

The main structure of the Hale Street Bridge, being constructed across the Brisbane river, is a parabolic arched bridge with a span of 60 meters. The maximum height of the arch above water level must ...
2
votes
1answer
33 views

is the sum of roots of a quadratic with rational coefficients always rational

quadratic is $ax^2 + bx +c = 0$ let the roots be $f$ and $g$ as $f + g = -\frac{b}{a}\ $ and $\ f \cdot g = \frac{c}{a}$ does this imply if a quadratic has rational coefficients the sum of the ...
6
votes
1answer
87 views

Coincidence? : $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$

As the title says, is it just a coincidence that $d(ax^2+bx+c)/dx=\pm \sqrt{\Delta}$? (where $\Delta=b^2-4ac$, i.e. discriminant of the quadratic). We can get this easily from rearranging the ...
0
votes
1answer
22 views

Can you simplify a coordinate

Can you simplify $(-6,-36)$ to $(-1,-6)$ if the first coordinate is the min value of a quadratic graph. The equation is $y = x^2 + 12x$.
0
votes
1answer
67 views

Solving an equation involving factorial notation

I was given this problem in the text book: $$\frac{(n+4)!}{(n+2)!} = 6$$ $$n \in I $$ Since the textbook doesn't have the solution, I'm wondering if I'm right: $$\frac{(n+4)!}{(n+2)!} \Rightarrow ...
1
vote
1answer
27 views

Sum of $n^{\text{th}}$ powers of roots of quadratic

How would I go about finding an expression (preferably closed form) for the sum of $\alpha^n+\beta^n$ in terms of $\alpha + \beta$ and $\alpha\beta$ (where $\alpha$ and $\beta$ are the roots of a ...
1
vote
1answer
40 views

Quadratic Functions Word Problem

A holding pen is being built alongside a long building. The pen requires only three fenced sides, with the building forming the fourth side. There is enough material for 90m of fencing. Predict what ...
0
votes
1answer
37 views

solve an quadratic equation

I was reading a document , where I stucked in figuring out this equation. $f(k)= k^2-nk+\frac{n^2 - n}{2}$. This is a quadratic function of $k$. It is minimized when $k=\frac{n}{2}$ (the $k$ ...
2
votes
2answers
83 views

Find the value of $f(x)$ for $x = 2 + 2^{2/3} + 2^{1/3}$

If $x = 2 + 2^{2/3} + 2^{1/3}$, then find the value of $f(x)=x^3 - 6x^2 + 6x$. I am unable to get to the answer - end up with more than one term. Please help me solve this!
1
vote
1answer
43 views

For the given quadratic equation find the value of p

For the equation 3x^2 + px + 3 = 0 , p>0, if one of the roots is square of the other, then p is equal to? Solving the equation, i get the value of p as -6 but the question states that p>0. Is there ...
0
votes
2answers
21 views

Quadratic Function: X intercepts.

A quadratic function with a y-intercept of 0 and an axis of symmetry of x=-1. Apparently, there is suppose to be 2 x-intercepts, which I really don't understand. How can the parabola cross the x ...