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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34 views

Quadratic equation form

I have the relation $u=\sqrt{(a_1+b_1t)^2+(a_2+b_2t)^2+(a_3+b_3t)^2} \tag 1$ I need to write $t$ as a function of $u$ ($t=f(u)$). How will I get that ? NB: $a_1,a_2,a_3,b_1,b_2,b_3$ are ...
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3answers
67 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 ...
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2answers
58 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?
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3answers
70 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 ...
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1answer
61 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 ...
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2answers
20 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 ...
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2answers
49 views

Solving cumbersome “quadratic” equation

Solving the equation of the form $$1-3x^2+3x\sqrt{1-2x^2}=0 $$ Is cumbersome since setting $t=1-2x^2$ does not yield an explicit quadratic formula in terms of t. There is some trix to this, but I ...
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2answers
59 views

How to find the number of solutions of equation $x^n - a^x = 0$?

I have to find the number of solutions of the equation $x^4 - 5^x = 0$ Since it is only asked to find the number of solutions and not the exact solution, what is the best way to approach such ...
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1answer
38 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 ...
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1answer
38 views

Wrong solution set in textbook, quadratic equation

So we have an the equation $\frac{2}{3}t^2+\frac{4}{3}t=\frac 15$, when you finish solving the equation you get $t = \frac{-10 + \sqrt{130}}{10} $ and $\frac{-10 - \sqrt{130}}{10}$. The text book ...
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2answers
37 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 ...
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1answer
26 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. ...
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1answer
37 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 $y$?
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3answers
30 views

Solving Quadratic equation by factorizing

The question is 2x^2 - 7x + 3 = 0. First of all quadratic equation is written in the form of ax^2 + bx + c = 0 in where a,b and c are numbers. In this equation I was told to use the matrix method. ...
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1answer
26 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: $$ xy + ky - x^2 - 3x = 0 $$ ...
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2answers
28 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 - ...
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1answer
110 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.
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1answer
69 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 ...
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2answers
51 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.
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25 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 ...
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1answer
27 views

Expand and simplify

I've removed the brackets from the first equation in where it is 5a^2 + 2a - 5 and then multiplied 3 to the numbers inside the bracket. But I'm not sure what steps to take after that. (5a^2 + 2a - 5) ...
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2answers
39 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...
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1answer
35 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 ...
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1answer
39 views

Deriving coefficient q in reduced quadratic equation using Vieta's formulas

I have the following example of the use of Vieta's formulas in my textbook: Let's have a quadratic equation of the form $x^2+x+q=0$ The following conditions apply: the equation has two roots, such ...
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0answers
48 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 ...
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2answers
52 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 ...
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1answer
24 views

Application of the Rational Roots Theorem

Let f(x)=3x$^3$ - 40x$^2$ + 97x + 10 a. Find a rational number r such that f(r) = 0. (Hint: Use the rational roots theorem to narrow down possibilities for r.) So, I figured this part out. write r ...
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0answers
31 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?
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4answers
37 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.
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1answer
167 views

Without solving the equation, determine the nature of its roots: $x^2 + ax + a^2 = 0$

I'm working through the book Core Maths for Advanced Level on my own, and, after solving the above problem, I'm not getting the same answer as the book. So, given: $$x^2 + ax + a^2 = 0$$ Using the ...
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2answers
36 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?
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1answer
57 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 ...
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1answer
101 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 ...
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2answers
183 views
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1answer
30 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 ...
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3answers
65 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?
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2answers
105 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 ...
3
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1answer
48 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 ...
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1answer
114 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 ...
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1answer
35 views

For which values of $n$, the real part of the $n$-th root of unity is a quadratic irrational?

For which values of $n$, the real part of the $n$-th root of unity is a quadratic irrational? That is, when is it a root of a quadratic polynomial with integer coefficients? I believe that the answer ...
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1answer
25 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 $y = x^2 + 12x$.
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1answer
237 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 ...
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1answer
32 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 ...
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2answers
29 views

Equation with two unknowns

I have this equation to solve and I solved it but don't know if the result is correct. $\begin{cases}2\pi r_1+2\pi r_2=24 & (1) \\ \pi r_1^2+\pi r_2^2=20 & (2)\end{cases}$ Equation $(1)$ ...
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1answer
44 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 ...
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1answer
40 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 $k$ ...
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2answers
86 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!
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1answer
169 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 ...
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2answers
22 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 ...
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2answers
45 views

Quadratic Functions word problem.

I have the quadratic function $$y=\frac{1}{294}(x-84)^2-24$$ that represents the shape of the cables of a certain bridge. I am suppose to be determining the vertical height of the cables above the ...