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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0
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4answers
75 views

Find $\frac{a^3}{a^6 + 1}$ given a is a root of a quadratic equation

My question is: If a is a root of the equation $x^2 - 3x + 1 = 0$, then find the value of $\frac{a^3}{a^6 + 1}$. So, I figured we can use the Sridharacharya ...
2
votes
3answers
82 views

Finding maximum value for a function

I was working on this question to find the following function's maximum value.Let $$y=f(x)={{(\sqrt{-3+4x-x^2}+4)}}^2 + (x-5)^2$$ where $$1 \le x \le 3$$.I have to find it's maximum value. I tried by ...
3
votes
1answer
32 views

Problem with system of equations

I wonder how to solve this system of equations: $\begin{cases} 2x^2+y^2=43\\2x^2+4xy=78\end{cases}$ when I subtract I have $y(4x-y)=35$ but I don't if it is good way to look for the solutions.
-1
votes
1answer
39 views

Finding the rule of a quadratic graph

I am trying to find the rule. Insofar: $y = a(x-b)^2 + c$ Turning point is $ (1,9) $ So $ b = 1 $ and $ c = 9 $ $y = a(-4-1)^2 + 9$ $-16 = a(-4-1)^2 + 9 $ $-25 = a (-4-1)^2 $ $-25 = a (-5)^2 ...
0
votes
2answers
64 views

Reducing quartic equations to quadratic

I'm trying to re-learn basic math/algebra, and I can't get passed one question concerned with reducing quartic equation to a quadratic: Find, correct to 3 significant figures, all the roots of the ...
0
votes
1answer
334 views

Solving a distance/time/speed problem using the quadratic formula.

"The distance between Toronto and Ottawa is 352.72 km. The speed on a road trip from Ottawa to Toronto was double of the return, and therefore the drive took 2 hours less. What was the speed on the ...
0
votes
2answers
55 views

Rearranging quadratic equations

I have an equation I'd like to rearrange that I'm having trouble with: The equation goes: $$r = 2ut - 0.333(2ut)^2$$ (on a computer I enter ...
0
votes
1answer
79 views

Solve a bit tricky system of equations

I want to solve the system for $x$, $y$ and $z$. Is there any smart trick to solve it? $$\begin{cases} 2a(ax+by)+2c(cx+dy)+2zx=0 \\ 2b(ax+by)+2d(cx+dy)+2zy=0 \\ x^2+y^2-1=0\end{cases}$$ $a,b,c,d \in ...
0
votes
1answer
94 views

9 rectangles have the same area as 20 squares

This is a fun little question that I encountered on a problem solving assessment: A small area is covered by 20 identical square tiles or 9 identical rectangular tiles. The length of the side of ...
2
votes
1answer
75 views

Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$.

Find the range of values of $p$ if $(\cos p -1)x^{2}+(\cos p)x+\sin p =0$ has real roots in the variable $x$. Restrict the values of $p$ in $[0,2\pi]$. The given equation has real roots if: $$\cos^2 ...
-2
votes
3answers
51 views

A question on quadratic equations.. Given below in the picture.

PLease also tell how u got to the answer as I want to know the way to solve further questions
2
votes
2answers
62 views

How to solve the following pair of equation.

The pair of equation I need to solve is $x^2+12x+y^2-4y=24$ $x^2-6x+y^2+8y=25$ I have no idea on how to do these kinds of problems (may be by elimination?)
1
vote
2answers
66 views

If $ax^2+bx+c=0$ and $2x^2 +3x+4=0$ have a common root where $a,b,c \in \Bbb N$,find least value of $a+b+c$

problem :If $ax^2+bx+c=0$ and $2x^2 +3x+4=0$ have a common root where $a,b,c \in \Bbb N$,find least value of $a+b+c$ Solution: Here $2x^2 +3x+4=0$ will give complex roots These roots will ...
1
vote
1answer
21 views

Consider the following simultaneous equations in $x$ and $y$…where $a$ is a real constant: $x+y+axy=a$,$x-2y-xy^2$

Consider the following simultaneous equations in $x$ and $y$: $$x+y+axy=a$$ $$x-2y-xy^2=0$$ where $a$ is a real constant. Show that these equations admit real solutions in $x$ and $y$. I could not ...
1
vote
2answers
36 views

Three variable systems if equations.

Given the quadratic function $y=x^2 + 4$ and the linear function $y=x + b$, determine all the possible values of $b$ that would result in a system if equations with two solutions, exactly one ...
1
vote
0answers
68 views

Quadratic equations with prime coefficients

I recently decided to go through old high school notebooks and I found something marginally interesting. I used to note down all kinds of things I came across, and I thought this might be useful for ...
-3
votes
1answer
89 views

To find ? in equation $a^2+?^2=c$

How can we solve for $?$ in the below given Equation: $$a^2+?^2=c$$ I don§t want to use Square or Square root as the the number can be in decimals.
0
votes
2answers
56 views

Complex numbers - Quadratic formula?

Let a and b be real numbers. The complex number 4 - 5i is a root of the quadratic $z^2 + (a + 8i) z + (-39 + bi) = 0$. What is the other root? I did a lot of work on hand and plugging this into the ...
0
votes
2answers
55 views

Query about a statement on the consequence of two quadratic equations having a common root

I have read an answer (in this site, but I lost the question number) saying something like the following:- If the quadratic equations F(x) = 0 and f(x) = 0 have a common root, then the quadratics are ...
3
votes
5answers
439 views

Condition for a common root in two given quadratic equations

If $a,\;b,\;c$ are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root if $\;\displaystyle\frac da,\;\frac eb,\;\frac fc$ are in: Arithmetic Progression ...
0
votes
2answers
25 views

Find quadratic equation based on 2 tangents

I would like to know a way to find an quadratic equation that had 2 given tangents: For example here is 2 tangents equations: y = 1/2 x y = 2 x + 2 and 2 abscisses x = 0 x = 3 Is there a ...
0
votes
3answers
51 views

Solving an equation involving $x^2$

I have come to a question with the equation: $$6 = x^2 -7x + 6.$$ The answer is $7$. How do I do I find the solution to a problem involving $x^2$?
0
votes
3answers
177 views

Solving a Quadratic Equation “Using a Table and a Graph”

I need to find $a\in \Bbb Z, 0\le a\lt10 : f(1 + \frac{a}{10}) = 0$ for a number of different quadratic functions, for example $f(x) = -x^2 + 4x - 3$, by "using a table and a graph". Can someone ...
0
votes
1answer
33 views

information content of a quadratic surd

how much information is required to construct the equation: $$ X^2 - 2=0 \; ? $$ suppose, in a spirit of seasonal festivity, we squander a few further bits, and pamper ourselves with the additional ...
0
votes
0answers
59 views

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then

If $3x^{2}-2(a-d)x+(a^{2}+2(b^{2}+c^{2})+d^{2})=2(ab+bc+cd)$, then $A.$ a,b,c,d are in G.P. $B.$ a,b,c,d are in H.P. $C.$ a,b,c,d are in A.P. $D.$ None of the above Tried writing the expression as a ...
2
votes
1answer
50 views

how to prove roots quadratics

the quadratic equation $3(k+2)x^2+(k+5)x+k=0$ has real roots show $(k-1)(11k+25) \geq 0 $ If $\Delta$ greater than $0$ it has real roots so, $$\Delta = (k+5)^2 - 4 \cdot (3(k+2))\cdot k$$ ...
0
votes
5answers
61 views

$x^2+y^2=1, 5x+12y+13=0$ Simultaneous Equations

Can someone solve this for me and show working out? I just can't do it and I don't know why I am getting x and y wrong. It will be very much appreciated. As basic as possible as well please.
0
votes
1answer
75 views

Finding two unknowns in two quadratic polynomials with only knowing the divisors

There are two quadratic polynomials (dividends). These two polynomials are divided by two different linear polynomials like $x+1$ (divisors). The remainders are known, but the quotients are unknown. ...
0
votes
1answer
50 views

How can I solve $\frac{2x}{\sqrt{1-x^2}}=0$

This is what I could come up with: $\dfrac{2x}{\sqrt{1-x^2}}=0$ $\left(\dfrac{2x}{\sqrt{1-x^2}}\right)^2=0^2$ $\dfrac{4x^2}{1-x^2}=0$ I can't go forward from this point because of that stupid ...
1
vote
0answers
14 views

proving there exist another basis of non-degenerate quadratic space (V,B) other than the given basis

If {$v_i$} is a basis of non-degenerate quadratic space ($V,B$) (finite), prove that there exists another basis {$w_i$} such that $$B(v_i,w_j)=1 (i=j)$$ $$or 0(i \neq j)$$ Sorry for the ugly text ...
1
vote
1answer
39 views

“Alternative factorising method” for quadratics not working

In class my teacher showed us an alternative method for factorising quadratics which are more awkward (i.e. the $a$ in $ax^2+bx+c$ is greater than 1). The method is: 1. Take your quadratic (e.g. ...
1
vote
0answers
56 views

How surfaces intersect in projective spaces

Consider this parametrization $$\phi:\mathbb{P}^1\longrightarrow\mathbb{P}^3$$ $$(t_0:t_1)\longmapsto (t_0^3: t_0^2t_1:t_0t_1^2:t_1^3)$$ Let $\mathcal{C}$ be the image of $\phi$. I've proved that ...
0
votes
1answer
21 views

Given a set D = {a+b•| a,b ∈ $\mathbb{R}$} and a made-up binary operation, in a quadratic equation.

Given a set D = {a+b•| a,b ∈ $\mathbb{R}$} And a made-up binary operation on D is defined as follows: (a+b•)(c+d•)= ac+(ad+bc)• For example, (2+3•)(-3+5•)= (-6+1•) You are not allowed to combine ...
2
votes
1answer
51 views

Mechanics - Homework

A stone is thrown vertically down from a high building with an initial velocity of $4\;\mathrm ms^{-1}$. Calculate the time required for the stone to travel $30\;\mathrm m$. So far I have tried using ...
1
vote
6answers
217 views

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root

Find $(a,b)$ such that in $x^2+ax+b$, whenever $v$ is a root, then $v^2 - 2$ is also a root $a,b$ are real numbers. Roots may or may not be real. In this question, the aim is to find values of and b ...
4
votes
3answers
69 views

Finding value of equation without solving for a quadratic equation

How do I go about solving this problem: If $α$ and $β$ are the roots of $x^2+2x-3=0$, without solving the equation, find the values of $α^6 +β^6$. In my thoughts: I commenced by expanding $(α ...
3
votes
1answer
29 views

Finding number of solutions.

How many solutions does this equation have $$2 \cos^2\left(\frac12 x \right) \sin^2 x = x^2+x-2$$ where $0 \lt x \le \displaystyle\frac \pi9?$ I observed that $2 \cos^2\left(\frac12x\right)$ can be ...
1
vote
1answer
25 views

Quadratic Equations - Mixed Roots

This is probably a silly question but why is it that when a quadratic equation has a single root it must be a repeated root. Why can't the second root be an imaginary root?
0
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3answers
32 views

Intersecting a curve at the x axis

$Y=0$, $Y=x^2-4x-4$ How do I find the points of intersection (that is, all $x$ on both curves $Y = 0$ (the $x$-axis) and $Y = x^2 - 4x - 4$)?
6
votes
3answers
279 views

A new way of solving cubics?

I found this (from http://www.quora.com/Mathematics/What-are-some-interesting-lesser-known-uses-of-the-quadratic-formula): So my question is: Can this be generalized to solve any depressed cubic ...
0
votes
1answer
49 views

if f(x) is the polynomial (coeff of leadin term is unity) in 'x' of least degree such that f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1, then f(0)=?

If $f(x)$ is the polynomial (coefficient of leading term is unity) in 'x' of least degree such that $f(1)=5 , f(2)=4, f(3)=3, f(4)=2, f(5)=1$ Then $f(0)= ?$
2
votes
2answers
93 views

How to prove that if $-1<x<0$ then $x^2 + x < 0$?

I am trying to prove an equivalence. I have already proved that: $$x^2 + x < 0 \implies -1 < x < 0 $$ using a sub-proof by cases, in which I used the fact that when $xy < 0$, $x$ and ...
0
votes
2answers
148 views

How do I determine the maximum value for a quadratic equation on an interval?

I need to determine the maximum value for y = ax^2 + bx + c, where I know the coefficients and the upper and lower x values. Say the input values are: a = 5 b = ...
0
votes
1answer
19 views

absolute value in a quadratic

If $a<-2$ is a real number, then the equation: $x^2+a|x|+1=0$ has how many real roots? After finding the roots in terms of $a$, how do I proceed?
0
votes
1answer
69 views

Finding a+b+c+d where a,b and c,d are the roots of two different quadratic equations

If $a, b$ are the roots of the equation $x^2-10cx-11d=0$ and $c,d$ are the roots of the equation $x^2-10ax-11b=0$ (where $a\ne b\ne c\ne d\ne 0$), then find the value of $a+b+c+d$. I have the ...
0
votes
0answers
34 views

Complete the square

How would I complete the square for $y$ after completing the square for x below. Note that $y,x$ are vectors and not scalars. $$ (y-A^Tx)^TC(y-A^Tx)+(x-\mu)^TD(x-\mu)\\ ...
-1
votes
1answer
34 views

Quadratic Equations and graphs [closed]

A bridge forms a parabolic arch. The span of the arch is 80 meters and its centre is 15 meters above either end. Write a quadratic equation that models the arch.
0
votes
1answer
44 views

Modular quadratic equation (solve for 3-digit natural numbers)

$n^2 + 6n - 88$ is divisible by 97. Solve for all n if n is a 3-digit natural number. Here's my progress so far $$n^2 + 6n - 88 \equiv 0\pmod {97}$$ $$n^2 + 6n - 88 + 97 \equiv 0\pmod {97}$$ $$n^2 + ...
3
votes
0answers
145 views

Second longest prime diagonal in the Ulam spiral?

Given the Ulam spiral with center $C = 41$ and the numbers in a clockwise direction, we have, $$\begin{array}{cccccc} \color{red}{61}&62&63&64&\to\\ ...
0
votes
1answer
52 views

Algebra formulas: answer is positive, but in calculator it's negative.

$$-X^2 + 11X - 30 = 0 $$ $$\frac{-11 + \sqrt{11^2 -4 * 1*30}}{2*1} => \frac{-11 + \sqrt{1}}{2} => -5$$ Why do I get minus? In the book, it shows 5, not -5?