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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2answers
99 views

jenny farm and the dozen egg ???

Farmer Jenny decides to expand her business interests and starts to package and sell the eggs produced by her chooks to a local shop. The cost of producing $x$ dozen eggs per day is given by, in ...
2
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4answers
526 views

Solving a quadratic equation with precision when using floating point variables

I know how to solve a basic quadratic equation with the formula $t_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ but I learned that if $b\approx\sqrt{b^2-4ac}$ floating point precision may give slightly ...
3
votes
3answers
129 views

quadratic equation

If $\alpha$ is root of equation $x^2+x+1 = 0$ then find the value of $1+\alpha +\alpha^2+\alpha^3+\cdots+\alpha^{2010}$ Here I have put the value of $\alpha$ in the given equation to get $1+\alpha + ...
2
votes
2answers
344 views

Quadratic Equation relation between roots

If the ratio of the roots of the equation $x^2+px+q=0$ are equal to the ratio of the roots of the equation $x^2+bx+c=0$ , then prove that $p^2c=b^2q$ Let $\alpha \& \beta$ be the roots of first ...
2
votes
1answer
343 views

Quadratic Equation using surds property

$$\left(\sqrt{2+\sqrt{3}}\right)^x+\left(\sqrt{2-\sqrt{3}}\right)^x=2^x$$ Using property of surd can we simplify the above expression like: $$\left(\frac{\sqrt{3}+1}{\sqrt{2}}\right)^x ...
2
votes
1answer
57 views

Least value of $a$ for which at least one solution exists?

What is the least value of $a$ for which $$\frac{4}{\sin(x)}+\frac{1}{1-\sin(x)}=a$$ has atleast one solution in the interval $(0,\frac{\pi}{2})$? I first calculate $f'(x)$ and put it equal to $0$ to ...
4
votes
3answers
145 views

Values of $a$ for which $(a+4)x^2-2ax+2a-6 <0$ for all $x \in R$

How can we find all values of $a$ for which the inequality $(a+4)x^2-2ax+2a-6 <0$ is satisfied for all $x \in R$? For the given condition, $D >0$, therefore $ (-2a)^2-4(2a-6)(a+4) >0$. ...
1
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1answer
87 views

quadratic equation - nature of roots

For what values of a does the equation $$x^2-( 2^a-1)x-3(4^{a-1}2^{a-2})=0$$ possess real roots? Since the roots are to be real that means the discriminant should be $\geq 0$ $$\Rightarrow ...
0
votes
1answer
98 views

When finding the dilation factor of $y = 3(2x - 3)^2 - \frac{1}{4}$, why must the brackets be expanded?

When finding the dilation factor of $y = 3(2x - 3)^2 - \frac{1}{4}$, why must the brackets be expanded? Why can't the outside factor of $3$ simply be used for the dilation factor from the ...
3
votes
3answers
125 views

Equation in the real world

Does a quadratic equation like $x^2 - ax + y = 0$ describe anything in the real world? (I want to know, if there is something in the same way that $x^2$ is describing a square.)
3
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0answers
79 views

Question about linearization

Given a data matrix $D\in\mathbb{R}^{N \times N}$ Can one construct another matrix $M$ that for all permutation matrices $Q^A$,$Q^B$, if $[\sum_i\sum_j (Q^A_{ij}D_{ij})]^2 \geq [\sum_i\sum_j ...
5
votes
2answers
127 views

Find the value of $x_1^6 +x_2^6$ of this quadratic equation without solving it

I got this question for homework and I've never seen anything similar to it. Solve for $x_1^6+x_2^6$ for the following quadratic equation where $x_1$ and $x_2$ are the two real roots and $x_1 > ...
0
votes
1answer
150 views

Finding descent direction of quadratic function

I have a quadratic function: $f(x) = 24x_1+14x_2+x_1x_2$ and point $x_0 = (2,10)^T$ with $f(x_0) = 208$ And the first question is "give descent direction r in $x_0$" The second question "is f convex ...
2
votes
2answers
158 views

Solving for the length of a side of a triangle

I have a problem in which I'm supposed to solve for the length of the two sides of the triangle below. I assumed that it would simply boil down to $x+5=\sqrt{4x+52}$, and converted to standard form, ...
2
votes
2answers
1k views

Find value of $k$ for which the equation has real roots

What can be the value of $k$ for which the equation $9x^2+2kx-1=0$ has real roots? Things should be known When the quadratic equation has real roots, then $d=b^2-4ac \ge 0$ . Where ...
2
votes
3answers
1k views

If both roots of the Quadratic Equation are similar then prove that

If both roots of the equation $(a-b)x^2+(b-c)x+(c-a)=0$ are equal, prove that $2a=b+c$. Things should be known: Roots of a Quadratic Equations can be identified by: The roots can be ...
2
votes
4answers
893 views

Difference between fields $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$ and $\mathbb{Q}[\sqrt{2},\sqrt{3}]$? [duplicate]

Possible Duplicate: Is $\mathbf{Q}(\sqrt{2}, \sqrt{3}) = \mathbf{Q}(\sqrt{2}+\sqrt{3})$? How would one describe elements from $\mathbb{Q}[\sqrt{2}+\sqrt{3}]$ and ...
2
votes
2answers
73 views

Simple question regarding factoring quadratics

Say we have an equation $ax^2 + bx - c = 0$ and want to find $x$. Obviously the way to solve would be to use the quadratic equation or factorize. I understand that saying $$ax^2 + bx = c => x(ax + ...
2
votes
3answers
2k views

Modular Quadratic Formula

How can I solve quadratic equations using modular arithmetic? E.g. $$2x^2 + 8x + 2 = 0 \pmod{23}$$ N.b. I have changed the figures from those in my homework question because I don't want a solution ...
3
votes
1answer
66 views

Scale change on Quadratic

Consider the two functions $f(x)=ax^2$ and $g(x)=bx^2$. Using this transformation form $T(x,y)=(cx,cy)$, find a scale change that maps $f(x)$ onto $g(x)$
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3answers
95 views

How to solve systems of three equations?

Either I forgot or never did learn to do it well. I need to solve the following system: $$9a+3b+c=0$$ $$25a-5b+c=0$$ $$a-b+c=12$$ Google shows me this page with some instructions: ...
1
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2answers
251 views

Solving a quadratic equation via a tangent half-angle formula

(Maybe I'll post my own answer here, but maybe others will make that redundant.) This is a fun (?) trivia item that fell out of a bit of geometry I was thinking about. One of the tangent half-angle ...
4
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1answer
41 views

$(a - 1)x^2+3(a + 1)x+4(a - 1) = 0$ has real solutions iff $7a^2 - 50a + 7\leq 0 $

How can we show that $(a - 1)x^2+3(a + 1)x+4(a - 1) = 0$ has real solutions if and only if $7a^2 - 50a + 7\leq 0$? I know these are quadratics and can solve them, but I'm not entirely sure what the ...
4
votes
1answer
614 views

Relationship Between Roots and Coefficients of a Quadratic

To prove this lemma I use the relationship between roots and coefficients of a quadratic equation but did not get the result. Please help me prove this lemma. If ‎‎ $ - ‎\theta‎‎_{2}x^2 - ‎ ...
3
votes
1answer
337 views

Finding coefficients of quadratic given one tangent and point on the curve

I am given a quadratic equation: $$ y = Ax^2 + Bx + C $$ that passes through $(1,3)$ and $(2,3)$, and a tangent to the curve is $x - y + 1 = 0$ at $(2.3)$. How do I find $A$, $B$, and $C$? The ...
3
votes
1answer
58 views

Factorising a quadratic equation

I've just started studying for an A-Level in Mathematics. This is probably a simple question but when I factorized the quadratic equation $15x^2+42x-9$ I took out the common factor $3$ to get ...
3
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2answers
94 views

What is the Logic to be used for Solving this Problem?

I came across the following in a quiz contest qualification test: $$x = 2 + {1\over 2+ {\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}}$$ Find the value of: $$\frac{3x^2+5x -3}{2x^2 -4x+5}$$ Now, I know ...
0
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1answer
121 views

Identifying Quadratic equations from collected information

A girl can row her boat at $5 km/h$ in still water. If she takes $1$ hour more to row the boat $5.2 km$ upstream then to return downstream, find the speed of the stream. What I had done so far: Let, ...
0
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1answer
565 views

Right Triangle Hypotenuse in a right triangle (Quadratic Equation)

The hypotenuse of a right triangle is $5 m$ if the smaller is doubles and longer is triples the new hypotenuse is $6\sqrt{5} m$. FInd the sides of the triangle. What I found so far: After coming up ...
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2answers
120 views

Quadratic Equation Problem

A car covers distance of 1592cm. The number of hours taken for the journey is 1 half the number representing the speed in km/h. Find the time taken to cover distance. Hint: We will have to use the ...
2
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5answers
151 views

Show $x^2 +xy-y^2 = 0$ is only true when $x$ & $y$ are zero.

Show that it is impossible to find non-zero integers $x$ and $y$ satisfying $x^2 +xy-y^2 = 0$.
0
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1answer
55 views

When is the given function positive

I have to find the value of $x$ for which the given function is positive \begin{align} \alpha +\beta x + \sqrt{ax^2+bx+c} \end{align} I know that $ax^2+bx+c$ is always positive. Given conditions, ...
1
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1answer
73 views

Finding the fixed point

Trying to solve this question, got this answer but have a gut feeling that this might not be the way to do it, by the way this topic is related to fixedpoints The solution that I came up with
3
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2answers
276 views

How do you find the vertex of a (Bézier) quadratic curve?

Before I elaborate, I do not mean a quadratic function! I mean a quadratic curve as seen here. With these curves, you are given 3 points: the starting point, the control point, and the ending point. I ...
4
votes
3answers
92 views

Find the number of values of $a$?

Consider a quadratic equation; $$ x^2 + 7x – 14(a^2 + 1) = 0,$$ … (where $a$ is an integer) For how many different value of $a$, the equation will have at least one integer root? I found out its ...
7
votes
6answers
355 views

Solve $5a^2 - 4ab - b^2 + 9 = 0$, $ - 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0$

Solve $\left\{\begin{matrix} 5a^2 - 4ab - b^2 + 9 = 0\\ - 21a^2 - 10ab + 40a - b^2 + 8b - 12 = 0. \end{matrix}\right.$ I know that we can use quadratic equation twice, but then we'll get some ...
0
votes
1answer
200 views

An airplane makes a 990 km flight with a tailwind and returns, flying into the same wind.

An airplane makes a 990 km flight with a tailwind and returns, flying into the same wind.The total flying time is 3 hrs 20 mins and the airplanes speed in still air is 600 km/h what is the speed of ...
2
votes
6answers
242 views

Solving for x with exponents (algebra)

So I am trying to help a friend do her homework and I am a bit stuck. $$8x+3 = 3x^2$$ I can look at this and see that the answer is $3$, but I am having a hard time remembering how to solve for $x$ ...
4
votes
2answers
127 views

Finding the Extrema of a Function (without differetiation)

$$ (t^2-t+1)/(t^2+t+1) $$ prove that the function is upper bounded by 3 and lower bounded by 1/3 without differentiation
1
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1answer
78 views

How to find sum of quadratic

I got this quadratic function from physics that I need to find the sum of each term, up to whatever point. Written thusly: $$ \sum_{n=1}^{t}4.945n^2$$ And is there someway to quickly figure this ...
2
votes
3answers
4k views

Taking the square roots in inequalities

I have a question regarding taking square roots in inequalities. I have a problem asking: Suppose $3x^2+bx+7>0$ for every real number x. Show that $|b|<2\sqrt{21}$. In an earlier question it ...
1
vote
4answers
302 views

Is it possible to take the absolute value of both sides of an equation?

I have a problem that says: Suppose $3x^2+bx+7 > 0$ for every number $x$, Show that $|b|<2\sqrt21$. Since the quadratic is greater than 0, I assume that there are no real solutions since $y = ...
4
votes
4answers
395 views

Quadratic function concepts

My teacher was explaining quadratics in my class and it was a little bit unclear to me. The problem was Suppose $at^2 + 5t + 4 > 0$, show that $a > 25/16$ . My teacher said that there are ...
1
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1answer
3k views

Completing the square with negative x coefficients

I know how to complete the square with positive $x$ coefficients but how do you complete the square with negative $x$ coefficients? For example: \begin{align*} f(x) & = x^2 + 6x + 11 \\ & = ...
1
vote
1answer
63 views

What technique reduces factorable ax^2+bx+c=0 to factorable where a=1

The normal way to factor ax^2+bx+c=0 is to look for t,u,v,w such that: (tx+u)(vx+w) = 0 so that tv=a, uw=c, and uv+wt=b. This can be tricky, since there can be several possibilities for t,u,v,w. ...
3
votes
2answers
248 views

Quadratic equation to calculate a temperature from resistance

I'm trying to implement an electronic temperature sensor that gives a resistance value. The sensor is a Honeywell TD4. In the datasheet, they give a table of values : -40ºC => 1584Ω ±12Ω -30ºC => ...
2
votes
3answers
640 views

How was the quadratic formula found and proven? [duplicate]

Possible Duplicate: Why can ALL quadratic equations be solved by the quadratic formula? History of Quadratic Formula How was the quadratic formula $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ found ...
3
votes
4answers
234 views

Where is the order of the variables inside the parentheses coming from?

I'm reading Sawyer's Prelude to Mathematics, here: I can't understand what's the meaning and application of "condition" here. Also when he gives the example on the cubic equation, stating that ...
1
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2answers
1k views

Mathematics behind intersection points of two lines using quadratic equation

This is the question I am trying to solve. I do not need any code examples just help on mathematics. Suppose two line segments intersect. The two endpoints for the first line segment are $(x_1, ...
5
votes
5answers
4k views

Why a quadratic equations always equals zero?

On evaluating quadratic equations, It always equals zero: $$ax^2+bx+c=0$$ Why zero? Is it possible to use other number for another purpose?