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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1answer
92 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
72 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
105 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
121 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
1answer
113 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
114 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
80 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
69 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
269 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
107 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
112 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
51 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
160 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
95 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
322 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
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
votes
4answers
549 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
130 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
355 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
353 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
58 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
148 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
vote
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
99 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
126 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
votes
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
155 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
160 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
2k 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
900 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
74 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 $g(x)$
1
vote
3answers
98 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
vote
2answers
260 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
votes
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
622 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
359 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
votes
2answers
98 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
votes
1answer
124 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
votes
1answer
581 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
126 views

Quadratic Equation Problem [on hold]

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
votes
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
votes
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
vote
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
votes
2answers
280 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 ...