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
4answers
123 views

How do you solve $4x^2=-16x$? I get different answers depending on the method used.

I'm solving the following GRE problem: Solve $4x^2=-16x$ Method 1: I simply divide both sides by $4x$ :$$x=-4$$ Method 2: I solve by factoring:$$4x^2+16x=0$$ $$4x(x+4)=0$$ $$x=-4, x=0$$ Using ...
1
vote
3answers
160 views

Quadratic Equation find the value of $\lambda$ when other roots are given in restriction

Problem : If $\lambda$ be an integer and $\alpha, \beta$ be the roots of $4x^2-16x+\lambda$=0 such that $ 1 < \alpha <2$ and $2 < \beta <3$, then find the possible values of $\lambda$ ...
0
votes
0answers
158 views

Sample Code to Generate Points on the Rim of a Randomly Rotated Cone : What's Going On Here?

Related to this question: http://math.stackexchange.com/questions/407897/randomly-generate-point-on-shell-from-3-points-2-angles-with-uniform-angle-dis I'm trying to reverse engineer the ...
1
vote
2answers
72 views

Quadratic expression in $x$ with roots $\frac gh$ and $-\frac hg$

What quadratic expression in $x$ has roots $$\frac{g}{h}\qquad\text{and}\qquad-\frac{h}{g}?$$ I know that this can be factored as $$\left ( x-\frac{g}{h} \right )\left ( x+\frac{h}{g} \right )=0$$ But ...
5
votes
2answers
86 views

System of Pythagorean Quadratics

I have a system of quadratics, obtained from three mechanical links, fixed at one end and free at the other. The intersection point of the three free ends is required. ...
1
vote
1answer
229 views

Finding the rational values of constant for which these constants are roots of equation

Problem : Determine all rational values for which $a,b,c$ are the roots of $x^3+ax^2+bx+c=0$ Solution : Sum of the roots $a+b+c = -a$ ........(i) ( Since , as per question $a,b,c$ are roots of ...
0
votes
2answers
38 views

System of equations in x and y

Solve for $x, y \in \mathbb{R} $ $$ 5x \left(1+\frac{1}{x^2+y^2}\right) =12$$ $$ 5y \left(1-\frac{1}{x^2+y^2}\right) =4$$ I need a Different Approach apart from what i posted..Thank You
3
votes
3answers
185 views

Is it possible to find out $x^2$ parabola and function from 3 given points?

I am programming a ball falling down from a cliff and bouncing back. The physics can be ignored and I want to use a simple $y = ax^2$ parabola to draw the falling ball. I have given two points, the ...
1
vote
4answers
67 views

Quadratic Formula problem?

There is a right triangle. The hypotenuse is 17 units. The sum of the other two sides is 23. Find the length of the two other sides. Thanks for everyone's help in advance!
1
vote
2answers
35 views

Maximum of d(12-d)

I'm a little confused on a quite simple quadratic problem. I need to calculate the maximum of $d(12-d)$ using basic quadratics. The answer is $6$ as can also be shown by $f'(x)= -2d +12$, however this ...
3
votes
1answer
216 views

Curve through four points — simple algebra??

The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to ...
3
votes
3answers
166 views

Proving the quadratic formula (for dummies) [duplicate]

I have looked at this question, and also at this one, but I don't understand how the quadratic formula can change from $ax^2+bx+c=0$ to $x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$. I am not particularly good ...
2
votes
1answer
158 views

Finding a prime $p$ to solve a quadratic congruence $\pmod{p}$

I have a congruence of the form $$ax^2+bx \equiv -1 \pmod{p},$$ where $p$ is an odd prime and $a,b \in \mathbb{Z}$. Given $a$ and $b$, is there a general method to finding $p$ such that the above ...
0
votes
1answer
97 views

Finding a polynomial of degree $n$ when value of $f(k)$ is equal to some value

Problem : If $f(x)$ is a polynomial of degree $n$ and if $f(k) = \frac{k}{k+1}$ where $k =0,1,2,\ldots,n$, find $f(x)$. Can we go like this : Let the polynomial be ...
3
votes
7answers
2k views

The perimeter of the rectangle is $20$, diagonal is $8$ and side is $x$. Show that $x^2-10x+18=0$

My friends recently took a Maths GCSE. In the paper, they came across a very difficult question which we spent a full half-hour train journey trying to figure out. We didn't manage it, so I've come ...
1
vote
1answer
116 views

Satisfying a condition on given quadratic equation

Let $P(x) = x^2 +2bx + c$ be a quadratic form where $b,c$ are real numbers.If $b^2 < c$ , show that $P(x) > 0$ for all $x$ .Is the converse also true? The value of $x$ after solving the ...
0
votes
2answers
61 views

Quadratic and geometric average

I'd like to find the find the quadratic average and the geometric average. To do this I have these informations : The standart deviation, the arithmetic average and the number of values. I know the ...
1
vote
1answer
91 views

what if geometric sequence + geometric sequence

I wrote a program that basicly can find the formula of the sequence that created with any-degree equation. For example if you give my program that sequence: [1926, 2811, 833240, 28778265, 398155842, ...
2
votes
1answer
311 views

Quadratic Equation Modulo an even composite

I am familiar with using the quadratic formula and Tonelli-Shanks with Hensel's Lifting Lemma to solve a quadratic equation, but how do I solve a quadratic equation in an even modulus? I can't use the ...
1
vote
6answers
1k views

Is a Quadratic equation a function?

The definition of a function is "A function is a relation in which there is never more then one value of the dependent variable for every value of the independent variable." Since a quadratic ...
3
votes
3answers
136 views

Solving quadratic equations by completing the square.

Graphing $y=ax^2+ bx + c$ by completing the square Add and subtract the square of half the coefficent of $x$. Group the perfect square trinomial. Write the trinomial as a square of a ...
2
votes
2answers
256 views

sum of squares of the roots of equation

The equation is $$x^2-7[x]+5=0.$$ Here $[x]$ the greatest integer less than or equal to $x$. Some other method other than brute forcing. I tried a method of putting $[x]=q$ and $x=q+r$ which gives an ...
1
vote
1answer
93 views

Application of quadratic functions to measurement and graphing

thanks for any help! Q1. Find the equation of the surface area function of a cylindrical grain silo. The input variable is the radius (r). (the equation is to be graphed using a graphics calculator ...
0
votes
2answers
107 views

Irreducibility of quadratic polynomial in Z[x]

I would like to ask, how to test irreducibility of quadratic polynomial. I found, that when square root of discriminant is integer, $\sqrt{D}\in Z, D=b^2-4ac$, the polynomial can reduced. The document ...
0
votes
2answers
251 views

Quadratic equation with tricky conditions. Need to prove resulting inequalities.

The roots of the quadratic equation $ax^ 2-bx+c=0,$ $a>0$, both lie within the interval $[2,\frac{12}{5}]$. Prove that: (a) $a \leq b \leq c <a+b$. (b) ...
1
vote
3answers
77 views

Why is the coefficient of $x$ in $\frac{1}{x}=0$?

I usually solve a quadratic equation: $$ax^2+bx+c=0$$ Through a method I learned in school: For a monic quadratic, you make $x=y-\frac{b}{2}$. The method is intended for a monic equation but in ...
5
votes
3answers
171 views

If $a+b=x$ and $ab=y$, what is the quickest way to solve for $a$ and $b$?

The mechanistic approach would be to simply substitute $b=y/a$ in the first equation to obtain a quadratic in $a$. But seeing the simplicity of the givens, I feel that there must be some better and ...
-1
votes
1answer
96 views

Quadratic Baseball Question

The height of a baseball is modeled by the function $h(x)=-0.005x^2+0.3x+1.5$, would an outfielder which is modeled by the function $m(x)=-0.06x+5.6$ where $50 \le x \le 90$, catch the ball?
-1
votes
3answers
73 views

Find the min value of $3a+b$

If $ax^2+bx+c=0$ has no real roots then find min value of $3a+b$ for $c=6$; Please tell me how to proceed , i don't have any clue of what to do.
6
votes
9answers
2k views

Prove $ax^2+bx+c=0$ has no rational roots if $a,b,c$ are odd

If $a,b,c$ are odd, how can we prove that $ax^2+bx+c=0$ has no rational roots? I was unable to proceed beyond this: Roots are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and rational numbers are of the form ...
4
votes
1answer
109 views

Necessary and sufficient conditions that the difference of two quadratic equations has no solutions in $\mathbb{N}$

Suppose you have an equation of the form $$ a(n^2 - m^2) + b(n-m) + c = 0 $$ With given integers $a$, $b$ and $c$. Is there a necessary and sufficient condition that the equation has no solutions ...
0
votes
1answer
140 views

Quadratic Equation - Nature of roots

What is the product of real roots of the equation $t^2x^2+|x|+q=0$ Since the complex equation is positive so sum of the roots are positive, here I am having four option as answers : $>0$ ...
3
votes
2answers
134 views

Convexity of Quadratic equation Inequality?

Solving an inequality of the form $x^TAx\geq0$ or $x^TAx\leq0$ is straightforward. I mean we have to check if A is positive semidefinite or negative semidefinite. But what would be the solution to the ...
3
votes
1answer
126 views

Question on quadratic problem set

Okay so I have a quadratic function problem. I will omit the problem for now just because we don't really need it. My problem is: M is surface area. Do I have to write M(x, y) or just M in the area ...
0
votes
1answer
88 views

Is this quadratic word problem correct so far?

I'm a little confused as to how to solve this word problem I have. The problem is: A rectangular box (with a top) has a square base. The sum of the lengths of its edges is 8 feet. What dimensions ...
1
vote
1answer
70 views

How to show $\frac{300}{v} - \frac{300}{v+20} = 1.25$

A man travels a distance of $300$ km. On his return journey his average speed increased by $20$ km/h and his journey time decreased by $1\frac{1}{4}$ hours. If $v$ is the average speed of his outward ...
-1
votes
3answers
286 views

Determine the of p and other roots. [closed]

One of the roots of $3x^2 + p =5x$, is $2$. Determine the value of $p$ and the other root.
1
vote
2answers
109 views

Linear Regression to quadratic function

What is the optimal linear regression (w and w/o y-intercept) for a quadratic curve w.r.t. mean square error. Mathematically speaking: Given, $$y = x^2$$ for $$x = [-a,a]$$. What is the best ...
2
votes
2answers
115 views

Help in understanding quadratic equation

Sorry if this is a complete dummy question, but I haven't done math in years and I'm quite rusty. I'm reading this explanation of least squares regression, which internally uses the quadratic equation ...
0
votes
2answers
52 views

is there an analytic solution to $n^2+kn-d=m^2$ m,n integers

For $k=24,d=-17;m=8,n=3$, completing the square gives $(12+n)^2=m^2+161$ Where $161$ just happens to be the product of two primes $(q=7,p=23)$, so for large $k,m,n$ factoring may be very slow. ...
1
vote
4answers
161 views

Simplifying $\{ x \in \mathbb{R} : x^2 - x - 6 \geq 0 \}$ when possible

Simplify the following interval notation when possible: $$\{ x \in \mathbb{R} : x^2 - x - 6 \geq 0 \}$$
1
vote
1answer
96 views

Find value of $k$

For what value of $k$, are the roots of the quadratic equation $$(k+4)x^2 + (k+1)x +1 = 0$$ equal.
0
votes
5answers
357 views

How to solve systems of equations with multiplication & addition.

So I have a system of equations: $$a + b = 12$$ $$a \cdot b = 36$$ In this case, $a$ and $b$ are both $6$, this can be easily done in your head. However, how can you scale this for larger problems?
1
vote
2answers
100 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
votes
4answers
640 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
132 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
363 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
421 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
63 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
153 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$. ...