Questions on quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

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1
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3answers
54 views

find total integer solutions for $(x-2)(x-10)=3^y$

I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard. How many integer solutions ($x$, $y$) are there of the ...
1
vote
4answers
75 views

Simultaneous equations, $\frac{1}{x}+\frac{1}{y}=1$,$x+y=a$,$\frac{y}{x}=m$

By eliminating $x$ and $y$ from the following equations, I need to find the relation between $m$ and $a$. \begin{align*} \frac{1}{x}+\frac{1}{y}=1 \\ x+y=a \\ \frac{y}{x}=m \end{align*} I tried ...
3
votes
1answer
80 views

reference on $\sqrt{ax}+\sqrt{by}=c$ as a parabola?

Does anyone have a reference on the equation $$\sqrt{ax}\,+\sqrt{by}=c\ ?$$ Clearing square roots and rearranging gives $$ax+by = \frac{(ax-by)^2+c^4}{2c^2}$$ This is the equation of a parabola, so ...
2
votes
2answers
39 views

Trigonometric equation cos sin and power

The problem is $2\cos t - 3\sin^2t +2 = 0$. I get to $2\cos t -3\sin^2t =-2$ I think that I need to use a trigonometric identity like $\cos(x+y)$ and to divide $2\cos t -3\sin^2t$ with the ...
0
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2answers
29 views

Nature of Roots of quadratic $af(x) = (x^2+2)(a-1)$

I need help with the following. The problem is stated like so: "The value of the constant $a$ is such that the quadratic function $f(x) \equiv x^2 +4x + a +3$ is never negative. Determine the nature ...
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votes
2answers
25 views

Find equation of a line intersecting parabola at one point only?

Find equation of a line with gradient equal to $2$ which intersects the parabola $y = 6 − x − x^2$ at one point? I tried using the equation of the line $y=2x+c$ and making it equal to $y= -x^2 - x ...
2
votes
1answer
18 views

Finding original price of Tea per kg

A reduction of $ 2 per kg enables a man to purchase 2 kg more tea for $8. Find original price of tea per kg Attempt Let price be $x per kg of tea .So let man buys 10 kg of tea .So total cost is ...
0
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2answers
23 views

Find k for which the equation has equal roots.

I find to find the value of k in terms of $\alpha$ and $\beta$ . I rearranged the equation $x^2 +kx-1=2k+x$ into $x^2+(k-1)x-(1+2k)=0.$ I then found $\alpha$$\beta$ to equal$-1-2k$. and $\alpha + ...
0
votes
3answers
27 views

Ranges of k values in quadratic for it to be positive

Here is the problem $2x^2+6x+1+k(x^2+2)$, find the condition that must be satisfied by k in order that the expression may be positive for all real values of x. The quadratic of the form $ax^2+bx+c$ ...
0
votes
0answers
15 views

Find a set of linear equations whose solution is the same as the minimum of a given quadratic objective function

Given an objective function $||x_1 a_1 - x_2 a_2 - b||$, where $a_1, a_2$ and $b$ are 3-dimensional vectors, how can I find the two linear equations for the $x_1$ and $x_2$ whose solution will find ...
-3
votes
3answers
30 views

Show the equation $x^2+(3a-2)x+a(a-1)=0$ has real roots for all values of a∈R and show that $x^2-x+1$ has same sign for all values of x [on hold]

How to show the equation $x^2+(3a-2)x+a(a-1)=0$ has real roots for all values of a∈R How to show that $x^2-x+1$ has the same sign for all values of x.
0
votes
2answers
38 views

How do I find a variable when there is a polynomial?

So I have two similar questions: $x=x_0 + v_0t + \frac{1}{2}at^2$ where I have to solve for $t$ and $mgx + \frac{1}{2}kx^2 = \frac{1}{2}mv^2$ where I have to solve for $x$ I'm not sure if I have ...
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votes
0answers
28 views

Area word problem [closed]

A rectangular swimming pool measures 10m by 4m. It is surrounded by a deck of uniform width. What is the width of the deck? Please provide steps.
0
votes
1answer
19 views

Possible value of $x$ so that fractions are in simplest form.

Which of the following could be the possible value of $x$ for which each of the fractions is in its simplest form, where $\lfloor{x\rfloor}$ stands for greatest integer less than or equal to ...
1
vote
3answers
41 views

Finding a sum involving roots of a quadratic equation

If $\alpha,\beta$ are roots of the equation $x^2-2x-7=0$ and $$S_r=\left(\frac{r}{\alpha ^r}+\frac{r}{\beta ^r}\right)$$ then find the value of $$\lim _{n \to \infty} \sum _{r=1} ^n S_r$$ I am unable ...
1
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2answers
39 views

$x^2 + (k-3)x + k = 0$, ranges of k for roots to be of same sign

I need some help on the following. The quadratic that I am dealing with is $x^2 + (k-3)x + k = 0$, and I need to find ranges of values of $k$, for which the roots will have the same sign. For the ...
0
votes
2answers
32 views

Find the greatest value of $b$.

If one of the roots of the equation $(a-b)x^2+ax+1=0$ is double of the other and is real , find the greatest value of $b$. $\color{green}{a.)\ \dfrac98} \quad \quad \quad \quad \quad b.)\ ...
2
votes
4answers
51 views

Using the complete the sqaure formula.. $2x^2 - 4x +1 = 0$

So Im using the complete square method and i was just wondering where am i going wrong. I'm solving this $2x^2 - 4x +1 = 0$ So i am using this rules. $$ax^2 + bx +c = 0$$ subract c from both ...
2
votes
3answers
36 views

Values of $p$ for which quadratic possess at least one positive root.

For what values of $p$ would the equation $x^2+2(p-1)x+(p+5)=0,\ \ \{x,p\}\in \mathbb{R}$ possess at least one positive root ? I tried $$[2(p-1)]^{2}-4(p+5)\geq 0\\~\\ \implies p\geq 4 \cup p\leq ...
2
votes
1answer
30 views

Find $p$ and $q$ in $x^2-px+q=0$

If $p$ and $q$ are the roots of the equation $x^2-px+q=0,\ \{x,p,q\}\in\mathbb{R} $, then find $p$ and $q$. I tried sum and product of the roots formula and got , $$p+q=p \\pq=q$$ I found $q=0$ ...
0
votes
1answer
41 views

Different answers of a quadratic equation.

given $4x^2−4x-5=0$ we all know the solution but what my teacher showed me is different after we get the \begin{align*} x & = \frac{4 \pm \sqrt{96}}{8}\\ x & = \frac{4 \pm \sqrt{4 \cdot ...
1
vote
2answers
26 views

Real Roots of Complex Quadratic Equation - (Kasana's first example)

I recently bought H.S. Kasana's Complex Variables. It seems quite interesting, and a little harder for me than I had expected, though I should be able to get through it if I take my time. ...
0
votes
1answer
17 views

Solution review

The variables $p,q,r$ and $s$ are correlated with each other with the following relationships $\dfrac{s^{0.5}}{p}=\dfrac{q}{r^2}$ .The ranges of values of $p,q$ and $r$ are respectively: ...
6
votes
2answers
167 views
+50

Find the number of sets of $(a,b,c)$ for $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}=\frac{29}{72}$

If $\dfrac{1}{a}+\dfrac{1}{b}+\dfrac{1}{c}=\dfrac{29}{72},\ \ c<b<a<60,\ \ \{a,b,c\}\in\mathbb{N} $. How many sets of $(a,b,c)$ exists ? Options $a.)\ 3 \quad \quad \quad \quad ...
2
votes
2answers
64 views

Number of solutions of $a^{3}+2^{a+1}=a^4$.

Find the number of solutions of the following equation $$a^{3}+2^{a+1}=a^4,\ \ 1\leq a\leq 99,\ \ a\in\mathbb{N}$$. I tried , $$a^{3}+2^{a+1}=a^4\\ 2^{a+1}=a^4-a^{3}\\ 2^{a+1}=a^{3}(a-1)\\ ...
0
votes
3answers
29 views

Finding chord length with Sum and products?

The line $x + y − 1 = 0$ intersects the circle $x^2 + y^2 = 13$ at $A(\alpha_1, \alpha_2)$ and $B(\beta_1, \beta_2)$. Without finding the coordinates of A and B, find the length of the chord AB. ...
2
votes
4answers
196 views

Nature of the roots of quadratic equation

Here is the problem that I need to prove: If $x$ is real and $\displaystyle{\ p = \frac{3(x^2+1)}{(2x-1)}}$, prove that $\ p^2-3(p+3) \geq 0$ Here is what I did: \begin{align*} p(2x-1)=3(x^2+1) \\ ...
7
votes
2answers
44 views

Find the values of $\cos(\alpha+\beta) $ if the roots of an equation are given in terms of tan

It is given that $ \tan\frac{\alpha}{2} $ and $ \tan\frac{\beta}{2} $ are the zeroes of the equation $ 8x^2-26x+15=0$ then find the value of $\cos(\alpha+\beta$). I attempted to solve this but I ...
0
votes
1answer
42 views

I am not getting roots of $x^4-4x^3+3x^2+2x-30=0$

Applying Descartes's method, I determined the interim equation as $y^4-3y^2-28=0$. Then I went on to treat this as a product of $(y^2+ky+m)(y^2-ky+n)$. Comparing coefficients of $y^2$, $y$ and ...
5
votes
5answers
78 views

Solve $2^{a+3}=4^{a+2}-48,\ a\in \mathbb{R}$

Solve $2^{a+3}=4^{a+2}-48,\ a\in \mathbb{R}$ I tried to simplify it , $2^{a+3}=4^{a+2}-48\\ 2^{a+3}=2^{2(a+2)}-2^4\cdot 3\\ 2^{2a}-2^{a-1}- 3=0\\ $ I don't know how to go from here. This ...
0
votes
0answers
12 views

Solving a quadratic vector/tensor equation arising from connected Markov chains

I have a discrete-time finite-state aperiodic irreducible Markov chain, which is composed of $m$ identical component sub-chains. With probability $1-\mu$, in each time step each of these chains ...
2
votes
3answers
38 views

Find the relation between $a,b $ and $c$ in quadratic equation.

If the roots of the equation $a(b-c)x^2+b(c-a)x+c(a-b)=0,\ \ \{a,b,c,x\}\in \mathbb{R}$ are equal, then $a,b,c$ are in Options $a.)\ AP\\ b.)\ GP\\ \color{green}{c.)\ HP}\\ d.)\ \text{cannot be ...
2
votes
3answers
94 views

This is equation is giving me issues $x^2 - 6x + 15 = 0 $

I was given this equation $x^2 - 6x + 15 = 0 $ I tried to look for numbers whose sum is big and product of ac and i could not find any. I tried using the quadratic formula ...
1
vote
3answers
39 views

(Discriminant) For which values of k will the equation g(x) = x + k have two real roots that are of opposite signs?

I am currently in Grade 12 and came across the following question in a past paper: $$g(x) = \frac{2}{x+1}+1$$ The question asks: For which values of k will the equation $g(x) = x + k$ have two real ...
4
votes
1answer
95 views

Vieta's Formula failed?

Find the value of $p$ if $p$ and $q$ are the roots of the equation, $x^2+px+q=0, \ \ \{x,p,q\}\in\ \mathbb{R}$ By using vieta's formula for sum and product of roots, $\begin{cases} p+q=-p ...
0
votes
3answers
60 views
0
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2answers
34 views

Question on Quadratic equation

Q- If roots of quad. Equation $x^2-2ax+a^2+a-3=0$ are real and less than $3$ then, a) $a<2$ b)$2<a<3$ c)$a>4$ In this ques., i used $\frac{-b\pm\sqrt{b^2-4ac}}{2a}$ and then if $a$ ...
1
vote
0answers
16 views

Finding point closest to origin on a hyperboloid

(1) Let A be 3x3 real symmetric matrix. The eigenvalues of $A$ are $\lambda_1 = -6, \lambda_2 = 1, \lambda_3=4$ $q(x_1,x_2,x_3) = -x_1^2 + x_2^2 -x^2_3 + 10x_1x_3 = 1$. $A$ is the matrix of $q$. I ...
0
votes
3answers
45 views

Quadratic equation - What is the value of x?

"Find out the value of x by this equation: $(x+a)^2 = (2a-3x)^2$". (The answer should by the way, according to my book, be $x1 = 0.25a$ $x2 = 1.5a$ Here's how far I've gotten: $(x+a)^2 = ...
0
votes
2answers
53 views

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$

Solve $1<\left(\dfrac{3x^2-7x+8}{x^2+1}\right)\leq 2,\ \ x\in\mathbb{R}$ options $a.)\ 1<x<6\\ b.)\ 1 \leq x<6\\ c.)\ 1<x\leq 6\\ \color{green}{d.)\ 1\leq x \leq 6}$ I ...
1
vote
1answer
51 views

Find Quadratic Bezier curve equation based on its control points

If the 3 control points of the quadratic Bézier curve are known, how do you calculate algebraically the equation of that curve (which is an y=f(x) function)? Let's say I have.. P0 (x,y) - startPoint ...
2
votes
5answers
36 views

solve $\dfrac{x^2-|x|-12}{x-3}\geq 2x,\ \ x\in\mathbb{R}$.

solve $\dfrac{x^2-|x|-12}{x-3}\geq 2x,\ \ x\in\mathbb{R}$. options $a.)\ -101<x<25\\ b.)\ [-\infty,3]\\ c.)\ x\leq 3\\ \color{green}{d.)\ x<3}\\ $ I tried , Case $1$ ,for $ ...
0
votes
3answers
42 views

Using Discriminant to find equation of a line?

Find the equation of the tangent to the parabola $$y = x^2 − 5x − 3$$ that is parallel to the line $3x − y − 7 = 0$. I know how to solve this question utilizing differentiation, but I can't think of ...
0
votes
6answers
104 views

How to solve $\sqrt{2-x} = x$

How I see it: $$(\sqrt{2-x})^2 = x^2 $$ $$2-x = x^2 \implies x^2 + x - 2 = 0$$ $$x^2 + x - 2 = (x+2)(x-1)$$ So the solutions for $x$ are $-2$ or $1$, but my textbook says $1$ is the only answer. ...
0
votes
1answer
23 views

Second Degree Equations

I am having problems figuring out how to solve the following second degree equations: 2x$^2$ + 3x + 1 = 0 I can't get factors that add together to get 3 or multiply together to get 1: (2x + ?)(x ...
0
votes
2answers
48 views

Showing that the roots of the quadratic are real

If $x^2+bx+c=0$ has real roots, show that the roots of the equation $x^2+bx+c(x+a)(2x+b)=0$ are real for all real values of $a$. I could do it by standard way by proving determinant is postive. ...
-4
votes
6answers
113 views

Complete the square on $x^2+10x+9=0$ [closed]

Complete the square on the following quadratic $$ x^2+10x+9=0 $$ What value is added to both sides of the equation?
0
votes
2answers
54 views

Relation between coefficients of a quadratic if one root is the square of the other.

If one root of the equation $ax^2+bx+c=0$ is the square of the other prove that $b^3+ac(c+a)=3abc$ I couldn't understand how to start the problem I considered the two roots as $p$ and ...
1
vote
2answers
26 views

State the coordinates of the vertex and the number of $x$-intercepts for the following function

State the coordinates of the vertex and the number of $x$-intercepts for the following function: $$ y = -4x^2 + 1 $$ I am not really asking for a straight-up answer. If you could please tell ...
1
vote
1answer
45 views

Solving quartic equation using substitution

We are learning a lot about the history of our famous mathematicians and this specific one is stumping me. They want us to solve a problem a specific way and I can't seem to figure out how to do it. ...