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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1answer
18 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
190 views

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
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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
31 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
215 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
47 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
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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 ...
3
votes
5answers
87 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 ...
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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
39 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
97 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
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3answers
48 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
101 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 ...
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3answers
64 views
0
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2answers
35 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
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0answers
18 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
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3answers
48 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
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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
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1answer
70 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
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5answers
37 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
44 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 ...
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6answers
106 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
24 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 ...
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2answers
49 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. ...
0
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1answer
60 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
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1answer
48 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. ...
2
votes
0answers
74 views

Why the quadrature formula is exact one not an approximation?

I am reading this material on the algorithm of calculating the centroid of a polyhedron. I am confused by the last step of the deduction: The three coordinates of the centroid can be obtained: ...
0
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2answers
60 views

Question on proving quadratic inequality

Let $ax^2+bx+c$ = 0 be a quadratic equation and $\alpha$,$\beta$ are real roots. Condition for $\alpha < -1$ and $\beta > 1$. Show that $1 +\frac{c}{a}$ + $\left|\frac{b}{a}\right| < 0$. ...
2
votes
2answers
37 views

Quadratic Polynomial with complex coefficients

Let polynomial $p(z)=z^2+az+b$ be such that $a$and $b$ are complex numbers and $|p(z)|=1$ whenever $|z|=1$. Prove that $a=0$ and $b=0$. I could not make much progress. I let $z=e^{i\theta}$ and ...
0
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0answers
38 views

Let $a,b,c$ be the sides of triangle. No two of them are equal and $\lambda\in\Re$…

Problem : Let $a,b,c$ be the sides of triangle. No two of them are equal and $\lambda\in\Re$. If the roots of the equation $x^2+2(a+b+c)x+3\lambda(ab+bc+ca)=0$ are real, then find the range of ...
0
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2answers
46 views

$2x^2-16x+28$ into standard form

I think I'm just doing something stupid here, because I know it's not hard. Here's what I did: $$y-28+{\_\_\_}=2x^2-16+{\_\_\_}$$ $$y-28+{\_\_\_}=2(x^2-8+{\_\_\_})$$ $$y-28+16= 2(x^2-8+16)$$ ...
0
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4answers
67 views

Secondary solving method of polynomial

$$x+1+\frac{1}{x}=0$$ This is a fairly trivial and possibly bland equation to solve. But for the sake of the question I will display them here: $$x\left(x+1+\frac{1}{x}\right)=x(0)$$ $$x^2+x+1=0$$ ...
1
vote
3answers
48 views

Do perfect square trinomials only have one root?

I apologize for the basic question, but I'm just now learning of perfect square trinomials in my math class. Google hasn't provided any relevant answers. Throughout all of the examples I have been ...
2
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3answers
343 views

Finding the roots of a different Quadratic equation from the roots of a Given Quadratic equation

The Question: If $\alpha$ and $\beta$ are the roots of the equation $ax^2+bx+c=0$... Then find the roots of the equation $ax^2-bx(x-1)+c(x-1)^2=0$ My Attempt: The new equation can be ...
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3answers
39 views

Find p and q for y(x)=x^2+px+q [closed]

Find $p$ and $q$ for $y(x)=x^2+px+q$ if the function has minimum equal to $-4$ for $x=1$ Can anyone try to solve this please?
1
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2answers
46 views

Solve x for a quadratic equation (not finding zeroes)

With a linear function $f(x)=5x+2=q$ can be solved for $x$ by rewriting it as $x=(q-2)/5$ While with a quadratic function $f(x)=5x^2+2x+2=q$ how would you solve for both x's on one side? So you ...
1
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1answer
72 views

Quadratic polynomials describe the diagonal lines in the Ulam-Spiral

I'm trying to understand why is it possible to describe every diagonal line in the Ulam-Spiral with an quadratic polynomial $$2n\cdot(2n+b)+a = 4n^2 + 2nb +a$$ for $a, b \in \mathbb{N}$ and $n \in ...
1
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2answers
68 views

solve $\sqrt{x+7}<x$ for $x\in \mathbb{R}$

solve $\sqrt{x+7}<x$ I tried $\sqrt{x+7}<x\\ x+7<x^2\\ x^2-x-7>0\\ x\in \left(-\infty, \dfrac{1-\sqrt{29}}{2}\right) \cup \left( \dfrac{1+\sqrt{29}}{2},+\infty\right) $ I m not ...
3
votes
3answers
40 views

solve $|x-6|>|x^2-5x+9|$

solve $|x-6|>|x^2-5x+9|,\ \ x\in \mathbb{R}$ I have done $4$ cases. $1.)\ x-6>x^2-5x+9\ \ ,\implies x\in \emptyset \\ 2.)\ x-6<x^2-5x+9\ \ ,\implies x\in \mathbb{R} \\ 3.)\ ...
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2answers
20 views

graph quadratic form and find the equation of asymptotes

So I had this quadratic form that need to be graphed showing both original and new axes. And I also need to find out the equation of asymptotes. $$ \left\{ \begin{aligned} ...
0
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3answers
78 views

Solve $3^{2x} -2 \cdot 3^{x+5} + 3^{10} = 0$ for $x$

Here's the question: Solve for $x$ in $$3^{2x} - 2 \cdot 3^{x+5} + 3^{10} = 0$$ I know that you have to factor something out, I'm just not sure what that something is. Thanks in advance
4
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2answers
49 views

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $

Solve $x^2-|5x-3|-x<2,\ \ x\in \mathbb{R} $ I tried $x^2-|5x-3|-x<2$ , case $1$ , $x^2-(5x-3)-x<2,\ x\geq 0 \\ x^2-6x+1<0 \\ 3-2\sqrt2 < 3+2\sqrt2 \\ 0.17<x<5.8\\ $ ...
2
votes
2answers
119 views

Epsilon-Delta proof of $\lim_{x\to 2} x^2=4$

I have seen and understand the delta-epsilon proof of the limit of $x^2$ for $x\to2$, such as explained here: https://www.youtube.com/watch?v=gLpQgWWXgMM Now I am wondering, is there also another ...
0
votes
1answer
26 views

Solving a quadratic equation with complex coefficient

Express $z4$=-$\sqrt{3}$+i in polar form. Hence solve the equation $Z^2$=$z4$ for $z$ a complex number. You may leave the answer in polar form. My answer: $z4$ in polar form is 2cis-30$^{\circ}$ and ...
2
votes
2answers
301 views

Proving the an expression is larger than a simplified quadratic

Let p and q be positive real numbers. Prove that $$ (p + 2)(q+2)(p+q) \ge 16pq $$ Any explanation/answer would be extremely helpful. Thanks : )
5
votes
4answers
627 views

Why are the coefficients always equal?

Take the equation $ax^{2} + bx + c = 3x^{2} + 4x + 53$. Why is it always true that $a = 3, b = 4$ and $c = 53$? I've seen many examples like this where the coefficients are equated, and was just ...
1
vote
1answer
31 views

Why does this hyperboloid change into a surface? [duplicate]

Given this equation $x^2+y^2+z^2+2xy+2xz+2yz-x-y-z=6$ and the corresponding quadric: If I rearrange the equation to $(x+y+z-3)(x+y+z+2)=0$ (which is equivalent), I get: So, which is the right ...
1
vote
3answers
71 views

When do variables cancel out?

Sometimes if I randomly combine different equation and try to solve for a variable, one of them will cancel out. Why? For example: $\displaystyle x^2 = 4y^2$ and $\displaystyle x = 2y + 1$ And solve ...
1
vote
2answers
49 views

What type of equation is this? How to solve it?

$$m^4+a^4=0$$ , the answer I obtained is $$0+i1,0-i1$$ but the answer is given as a/sqrt(2)-a/sqrt(2),a/sqrt(2)+a/sqrt(2)