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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2
votes
2answers
129 views

maximum using completing the square

Is it just me, or this problem does sound weird? The Parks Department is fencing a rectangular dog-run (a place for dogs to exercise) in a local park. It will be 7 yards longer than 5 times its ...
3
votes
4answers
60 views

How to search quadratic function

If a graph of the quadratic function $f(x)=ax^2+bx+c$, where $a$, $b$ and $c$ are constants. If this function vertex is $(13,−169)$ and the distance between the two intersection points with the ...
1
vote
3answers
388 views

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots.

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots. MY attempt: We can open and get a bi quadratic but that is two difficult to show that it has real roots.THere must be an ...
5
votes
3answers
785 views

How to solve problems involving roots. $\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$

How to solve problems involving roots. If we square them they may go to fourth degree.There must be some technique to solve this. $$\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$$
0
votes
1answer
49 views

The number of integral values of $a$ for which the inequality $3- |x-a |>x^2$ is satisfied by at least one negative $x$, must be equal to 6

The number of integral values of $a$ for which the inequality $3- |x-a |>x^2$ is satisfied by at least one negative $x$, must be equal to 6. I don't know how to solve this. Can you help?
4
votes
1answer
627 views

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ with $a, b, c\in \mathbb N$, prove that $\log_5 {abc}\geq2$. The equations I could form are: 1) $f(0)>0$ and ...
1
vote
2answers
74 views

The no. of values of k for which $(16x^2+12x+39) + k(9x^2 -2x +11)$ is perfect square is:

I wanted to know, how can i determine the no. of values of k for which $(16x^2+12x+39) + k(9x^2 -2x +11)$ is a perfect square.($x \in R$) I have tried, since $x$ is real the discriminant must be ...
4
votes
2answers
184 views

Find all values of a for which the equation $x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0$ possesses at least two distinct negative roots

Find all values of a for which the equation $$x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0 $$ possesses at least two distinct negative roots. I am able to prove that all roots would be negative .How to proceed ...
0
votes
0answers
302 views

If $ a+b+c=4 ; a^2=b^2+c=6 ; a^3+b^3+c^3=8 $ Then find the value of $a^4+b^4+c^4$

If $a+b+c=4$ $ a^2=b^2+c=6$ (this is not symmetric equation) $a^3+b^3+c^3=8$ Then find the value of $a^4+b^4+c^4$
3
votes
2answers
130 views

For what $x\in[0,2\pi]$ is $\sin x < \cos 2x$

What's the set of all solutions to the inequality $\sin x < \cos 2x$ for $x \in [0, 2\pi]$? I know the answer is $[0, \frac{\pi}{6}) \cup (\frac{5\pi}{6}, \frac{3\pi}{2}) \cup (\frac{3\pi}{2}, ...
1
vote
1answer
75 views

The least value of $4x^2-4ax +a^2-2a+2$ on $[0,2]$ is $3$. What is the integer part of $a$?

The least value of $4x^2-4ax +a^2-2a+2$ on $[0,2]$ is $3$. What is the integer part of $a$? We know that minimum value of a quadratic is $-\cfrac{b}{2a}$. We will get one condition from here and ...
0
votes
1answer
113 views

Quadratic Polynomial Question - Solving for a coefficient using the discriminant

This question has been troubling me: A parabola whose equation is of the form $y = Bx^2$ (where B is a constant) has the line $20x - y + 20 = 0$ as a tangent. Find $B$. The explanation says, ...
0
votes
1answer
217 views

Quadratic expression that generate primes

I recently learned that there exist quadratic expression that generate some primes and some of these equations generate more primes than others. In the following video, the person shows the following ...
2
votes
4answers
209 views

$a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0 $

Let $a,\,b,\,c,\,d$ be distinct real numbers and $a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0$. Then find the ...
1
vote
4answers
204 views

Find all real numbers such that $\sqrt {x-1/x } + \sqrt {1 - 1/x} = x$

Find all real numbers such that $$\sqrt{x-1/x} + \sqrt{1 - 1/x} = x$$ My attempt to the solution : I tried to square both sides and tried to remove the root but the equation became of 6th ...
3
votes
1answer
235 views

$y =f(x) =(ax^2 + bx +c)/(dx^2+ex+f)$ We have to find the conditions for this it takes all real values.

$$ y=f(x)=\frac{ax^2+bx+c}{dx^2+ex+f} $$ We have to find the conditions for this it takes all real values. MY solution One approach is to equate it to y and for a quadratic of x and put discriminant ...
0
votes
1answer
57 views

Find the integer solutions

What are the pairs $(A,N)$ where $A,N$ are integers such that the following equation is satisfied: $\large A=\frac{-6+\sqrt{144-12N^2}}{6}$ I know that we should have: $k^2=144-12N^2$ for some ...
1
vote
1answer
275 views

Show that that if $p,q,r,s$ are real numbers and $pr=2(q+s)$, then at least one of the eqns $x^2+px+q=0$ and $x^2+rx+s=0$ has real roots.

Show that that if $p,q,r,s$ are real numbers and $pr=2(q+s)$, then at least one of the eqns $x^2+px+q=0$ and $x^2+rx+s=0$ has real roots. My Attempt to the solution we know to have a real solution ...
1
vote
2answers
106 views

How many real roots for $ax^2 + 12x + c = 0$?

If $a$ and $c$ are integers and $2 < a < 8$ and $-1 < c$, how many equations of the form $$ax^2+12x+c=0$$ have real roots?
3
votes
3answers
183 views

Systems of Quadratic Equations Question

looking for help on this question. Solve the following systems of equations algebraically using the quadratic formula. $$\begin{align} y& =-x^2+2x+9\\ y& =-5x^2+10x+12\end{align}$$ Any help ...
0
votes
1answer
276 views

Finding the Equation of Parabola

Write the equation in the form $y=a(x-h)^{2}+k$ with zeros -4 and 8, and an optimal value of 18. I'm not sure what "optimal value" means first of all- I think it means that the maximum value has ...
0
votes
3answers
75 views

How to isolate $v_m$?

Note: I am not asking anything pertaining to the physics of this question; only the mathematics. The physics is just given as a context to the problem for those interested, as opposed to simply saying ...
5
votes
4answers
124 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
163 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
163 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
232 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
214 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
220 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
171 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
159 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
124 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
63 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
96 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
331 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
137 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
278 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
99 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
109 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
259 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
97 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?