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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3answers
115 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
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2answers
182 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 ...
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1answer
78 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 ...
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0answers
79 views

If $f(y)=ax^2+bx+c$, does this imply that $x=\frac{-b \pm \sqrt{b^2-4a[c-f(y)]}}{2a}$?

The equation $x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}$ is known as the quadratic formula and is the solution to the quadratic equation $ax^2+bx+c=0$. Sometimes I encounter equations such as $x=y^2-y$. Is ...
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2answers
94 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 ...
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2answers
206 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) ...
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3answers
74 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 ...
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3answers
164 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 ...
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1answer
74 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?
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3answers
70 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.
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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
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1answer
102 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 ...
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1answer
97 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 : $>0$ ...
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1answer
100 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
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1answer
101 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 ...
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1answer
68 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 ...
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1answer
66 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 ...
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3answers
243 views

Determine the of p and other roots.

One of the roots of $3x^2 + p =5x$, is $2$. Determine the value of $p$ and the other root.
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2answers
100 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 ...
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2answers
102 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 ...
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2answers
50 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. ...
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4answers
159 views

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

Simplify the following interval notation when possible: $$\{ x \in \mathbb{R} : x^2 - x - 6 \geq 0 \}$$
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1answer
93 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.
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2answers
97 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 ...
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4answers
458 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
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3answers
119 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
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2answers
311 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
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1answer
298 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 ...
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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 ...
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3answers
140 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$. ...
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1answer
85 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 ...
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1answer
85 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
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3answers
124 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.)
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0answers
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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 ...
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2answers
123 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 > ...
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1answer
139 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 ...
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2answers
149 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, ...
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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
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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
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4answers
840 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
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2answers
69 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 + ...
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3answers
1k 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 ...
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1answer
57 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
85 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: ...
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2answers
228 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 ...
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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
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1answer
593 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
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1answer
277 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
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1answer
54 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
93 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 ...