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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0
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1answer
13 views

How to solve for a & b in a parabolic equation?

Not a mathematician at all here, actually a programmer and this came up... I need some help on something that is pretty basic I'm guessing. Given the following formula how would I go about solving ...
0
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5answers
83 views

Factorizing quadratics mentally

How would it be possible to factorize quadratics mentally, for example the following one? $$2x^2+7x+3$$ Maybe even something like $$3x^2+22x+24$$
0
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1answer
13 views

Finding parameter for quadratic equation

Given $x^2 - 3ax + a^2 = 0$ and $$\frac{x_1^4-x_2^4}{\sqrt{5}x_1x_2} + x_1 + x_2 -20x_1x_2 - 4 = 0$$ Find $a$. The answer is $1$ ($a = 1$) I tried to present $x_1^4 - x_2^4$ as ...
2
votes
3answers
43 views

Why is the graph of a quadratic function a parabola?

I'm sorry for the stupid question, but it seems that extensive googling didn't yield an answer. I've learned about parabolas, and how the parabola is the curve that is equidistant from a point ...
1
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0answers
65 views

If $a+b+c = 0$ then the quadratic equation $3ax^{2}+2bx +c=0$ has atleast one root in _________?

If $a+b+c = 0$ then the quadratic equation $3ax^{2}+2bx +c=0$ has atleast one root in _________? Rolle's theorem states that if $f(a) = f(b)$ then there exists a $p \in [a,b]$ such that : $f'(p) = ...
6
votes
2answers
61 views

Quadratic equation with one root in $[0,1]$ and other root in $[1,\infty]$

Find the values of $a$ for which $x^2-ax+2=0$ has one root in $[0,1]$ and other root in $[1,\infty]$. The twoo rots are $$\frac{a\pm\sqrt{a^2-8}}{2}$$ The smaller root should be less than $1$. So ...
-1
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0answers
35 views

How to solve this equation to find the interest rate?

So the equation that I derived at is: $$ \frac{1000}{v^5} = 425(1+v+v^2+v^3) $$ Where v is: $$\frac{1}{1+r}$$ Where r is the interest rate. The answer apparently is 8.5761% but I have no idea how ...
1
vote
4answers
79 views

How would one solve the following equation?

This equation is giving me a hard time. $$e^x(x^2+2x+1)=2$$ Can you show me how to solve this problem algebraically or exactly? I managed to solve it using my calculator with one of its graph ...
2
votes
2answers
40 views

Quadratic expression problem involving sides of a triangle

If $a,b,c$ be the sides of a triangle where $a\neq b\neq c$ and $\lambda \in R$, if roots of the equation $x^{2} + 2\lgroup a+b+c\rgroup x + 3\lambda\lgroup ab+bc+ca\rgroup =0$ are real, then what is ...
1
vote
2answers
50 views

Pairs of Quadratic equations

If each pair of the three equations $$x^2 + P_1x + q_1 = 0$$ $$x^2 + P_2x + q_2 = 0$$ $$x^2 + P_3x + q_3 = 0$$ have a common root, then prove that $$P_1^2 + P_2^2 + P_3^2 + 4(q_1 + q_2 + q_3) = 2 ( ...
0
votes
1answer
43 views

How to graph a Quadratic equation.

I have an equation that goes: $0.0001x^2 - 0.22x + 197$. I'm not asking for the answer, but instead, how can I graph it without dealing with these insanely tough numbers.
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2answers
20 views

I need help with a radicals questions

The roots of the equation $ax^2+bx+c=0$ are in the ratio of $2:3$. Determine an expression for $b$ in the terms of $a$, and $c$. I need help solving the question, please help, and thanks!
2
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2answers
61 views

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in G.P, find $x$.

If $x$, $\{x\}$, $\lfloor x\rfloor$ are in Geometric Progression, find $x$; $x \neq 0$. Here, $\{x\}=x-\lfloor x\rfloor$ Some properties are pretty evident: $$0\leq \{x\} < 1 \tag{1}$$ ...
0
votes
2answers
22 views

Problem with graphical representation of quadratic equation.

I have an equation $x^2 -x =0$ I know, it has two solutions, $x=0,1$ If I plot it using a math software, it shows a straight line passing through $1$, perpendicular to x-axis. (I have plotted, ...
0
votes
1answer
14 views

Proof quadratic congruent equation has no solutions in $\mathbb{N}$

In computer science, quadratic probing is used in hash tables. We choose a $c_1$ and $c_2$ in the hash formula $h(k,i) = (h'(k) + c_1 i + c_2 i^2) \mod{m}$ where $h'(k) = k \mod{m}$ and $m$ is the ...
5
votes
0answers
77 views

Probability that the roots of a quadratic equation are real

Roots of the quadratic equation $x^2+5x+3=0$ are $4\sin^2\alpha+a$ and $4\cos^2\alpha+a$. Another quadratic equation is $x^2+px+q=0$ where $p,q\in\mathbb{N}$ and $p,q\in[1,10]$. Find the ...
0
votes
1answer
21 views

proving theorems about quadratic function

The theorems I want to prove are: $$\text {if } a + b + c = 0 \text{ then } x_1 = 1 \text { and } x_2= c/a$$ $$ \text{if } a - b + c = 0 \text{ then } x_1 = - 1 \text{ and } x_2 = -c/a $$ where ...
0
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1answer
29 views

How to solve nonlinear system generated from $(x_i - x_j)^2 \approx f_{ij}$

Looking for advice/help with solving a nonlinear system generated from the equation: $(x_i - x_j)^2 \approx f_{ij}$ where $\textbf(X) = (x_1,x_2,x_3,...,x_n)$, $f_{ij}$ are known, solve for ...
0
votes
1answer
17 views

If $y=-x^2+2x+4$ intersects $y=mx$ at $P$ & $ Q$. Determine the value $m$ so the mid point of $P$ & $Q$ is origin

If $y=-x^2+2x+4$ intersects $y=mx$ at Point $P$ and Point $Q$. Determine the value m so the midpoint of P and Q is the origin. I solved this question when I graphed it out, I wonder if there is a way ...
0
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0answers
29 views

Roots of $f(x)=0$ and $g(x)=0$ are imaginary

Let $f(x)=ax^2+bx+c$ and $g(x)=Ax^2+bx+D$, where $a$ and $A$ are non-zero and $a,b,c,A,D\in R$ Roots of $f(x)=0$ and $g(x)=0$ are imaginary, then which of the following may be correct: (A) ...
0
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1answer
32 views

Quadratic inequalities 1

The question is : $2x^2-3x-1 \leq 0$ I've got the answers but I keep getting them wrong. If someone could write out the steps it would be greatly appreciated.
1
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0answers
15 views

Given Cartesian points A and B, how to move A by n units towards B? [duplicate]

I have two points, for which I know Cartesian coordinates. I want to move one towards the other over specific distance. That's about the same as saying I want the distance between the points equal ...
0
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2answers
38 views

Trigonometric equation solving / 4th degree polynomial

Okay, so I was solving a free body diagram problem, no need to send it, but I found a very huge problem in doing so, in any way I tried solving , I either got to an equation $\sin(x-0.11)=4\sin(2x)$ ...
1
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2answers
43 views

Quadratic equation

Here's the question below - $x^2 - 6x + (p^2 - 6 )^2$ is a perfect square , write down the possible values of p . My thoughts : I thought of this expansion method - $(A-B)^2 = A^2 - 2 AB + B^2$ ...
1
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2answers
51 views

Sketching the graph from function

I am having some issues with sketching the following graph. Any assistance is appreciated. The function $g(x) = -x^2 + 6x + 4$ , $K \leq x \leq 6$ where $K$ is a constant. Also, $g^{-1}(x)= -(y+3)^2 ...
0
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0answers
30 views

A simple proof involving a quadratic equation.

Am I doing this proof right? I am asked to prove $$\exists z \in \mathbb{R} \forall x \in \mathbb{R}^{+}[\exists y \in \mathbb{R}(y-x=y/x)\iff x \ne z]$$ Proof: Letting $z = 0$ (I figured that $x ...
2
votes
2answers
105 views

How to prove that the roots of this equation are integers?

Let there be an equation $a^2 + 4ab + b^2 - 121 = 0$ where I want to prove that a,b are integers. Then I want to find whether there are integer values of $b$ for which $a$ is also an integer. Let us ...
3
votes
1answer
38 views

Total number of values of $a$ such that the equation $x^2+ax+a+1=0$ has integral roots

Find the total number of values of $a$ such that the equation $x^2+ax+a+1=0$ has integral roots The equation can also be written as $$\left(x+\frac{a}{2}\right)^2=\frac{a^2-4a-4}{4}$$ So $a$ is a ...
0
votes
1answer
36 views

When do I solve a quadratic expression by either factorising, completing the square or use the quadratic formula?

What are the rules and conditions as to which method I should use. I know how to use them, just not when I should.
1
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2answers
55 views

Existence of real roots for vector quadratic equations

I have a vector quadratic equation of the form $\boldsymbol{x}^{T} \boldsymbol{A} \boldsymbol{x} + \boldsymbol{x}^{T} \boldsymbol{b} + c = 0$ where $\boldsymbol{A}$ is symmetric and for my particular ...
-3
votes
4answers
65 views

Simplify in the form: $x^n + \frac{1}{x^n}$ [closed]

Simplify: $$\left(x^2 + \sqrt2 + \frac{1}{x^2}\right)\left(x^2 - \sqrt2 + \frac{1}{x^2}\right)$$ in the form $x^n + \frac{1}{x^n}$
2
votes
3answers
33 views

Indices and Bases: Solve “x”

Solve the equation $2^x - 3^{x-1}=-(x+2)^2$ How I got this question? I created this question so I know the answer. The answer is 5. But I have no idea how to solve it. Take note that I cannot do ...
0
votes
2answers
30 views

Solving one variable in terms of the another

Let $y = x^2 - 2x + 6$. Express $x$ in terms of $y$. This is my working: $$ x^2 - 2x = y - 6, \\ x(x-2)= y - 6. $$ From this point, I got stuck as I can't fully factorize the $x$ out as seen ...
0
votes
2answers
35 views

Inverse of $y=2x^2-12x+13$

I'm having a problem finding the inverse of $y=2x^2-12x+13$. At the end I get to the following: $$x=3 \pm \frac{\sqrt{40+8y}}{4}$$ As far as I know the answer is suppose to be $x= 3 \pm ...
0
votes
1answer
48 views

possible real solutions of the equations

What are the possible real solutions of the equations $$1000=v_1^2+4v_2^2,100=v_1+4v_2$$ Its a physics question but I thought its not necessary to post here . Thank you.
0
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2answers
47 views

I need help factoring quadratics.

I need help factoring $x^4-2x^2-3=0$. I do not know how to factor this disguised quadratic.
0
votes
4answers
62 views

I do not get this question at all. I need to prove the an equation has a minimum. Quadratics involved.

Prove that $f(x)= (x-a)^2+(x-b)^2$ has a minimum when $x= \frac{a+b}{2}$. (Prove not verify) I do not get this question whatsoever, please help me.
1
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2answers
67 views

Find the quadratic equation, given three points on the parabola

Given that $f(x)=a(x-b)^2+c$, and that $f(1)=f(5)=-1$ and $f(0)=-11$, find the values of $a,b$,and $c$. I found the axis of symmetry is $3$, because $f$ has the same value at $1$ and $5$, and ...
1
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0answers
32 views

2nd Order Differential Equation Limits

Consider the differential equation $$ay′′ + by′ + cy = g(t)$$ where $a > 0$, $c > 0$, and $g(t)$ is a continuous function on $\mathbb R$. (a) If $y(t)$ is a solution of the above equation ...
1
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0answers
20 views

Expression for sum of matrix quadratics

I'm stuck on an algebra problem. For $i=1, \dots, N$, let $Y_i \in \mathbb{R}^{r \times p}$, and $\hat{M} = \frac{1}{N} \sum_{i=1}^N Y_i$ be the matrix of the element-wise averages. Suppose $V \in ...
0
votes
0answers
32 views

What algorithms are applicable to solve a inequality constraint Quadratic Optimization?

Suppose that we have a quadratic optimization problem $$(QP) \qquad \min \lbrace\frac{1}{2}x^TQX+ q^TX\rbrace $$ s.t. $$AX=a;$$ $$BX\le b;$$ $$X \ge 0;$$ where $Q \in \mathbb{R}^{n \times n}$ ...
1
vote
2answers
53 views

Find all real solutions: $36x^3 + 6x^2 = 9x$

I am having trouble finding all the real solutions to this problem. I do not undertand how to solve it, and I have a test on Polynomials tomorrow. The problem is "Find all real solutions using your ...
1
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1answer
30 views

Decouple a system of quadratic equations

We have $n$ real variables, $x_1, \ldots, x_n$. We'd like to solve the following system of quadratic equations. \begin{cases} \displaystyle\phantom{_{+1}}x_1 \sum_{j=1}^n x_j z_j = f_1(x_1);\\ ...
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votes
1answer
18 views
1
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7answers
91 views

Solve $3 = -x^2+4x$ by factoring

I have $3 = -x^2 + 4x$ and I need to solve it by factoring. According to wolframalpha the solution is $x_1 = 1, x_2 = 3$. \begin{align*} 3 & = -x^2 + 4x\\ x^2-4x+3 & = 0 \end{align*} ...
0
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0answers
30 views

Can we multiply these equations?

Example 2. Form the equation whose roots are $0,\pm a, \frac{c}{b}.$ The equation has to be satisfied by $$x=0,x=a,x = -a, x = \frac{c}{b};$$ therefore it is ...
2
votes
0answers
34 views

What is relation between a particular root of two polynomials?

We have $$x^3+(m+n+p-1)x^2-((m+n)(1-p)+2p-1-mn)x-(p-1)(m-1)(n-1)=0$$ in which $m,n\ge2, p\ge1$ are natural numbers. All the three roots of this cubic are positive. Let $\lambda$ be the least of them. ...
1
vote
5answers
39 views

Principal roots in the derivation of the quadratic formula

The following is an abbreviation of a common derivation of the quadratic formula: $$ ax^2+bx+c=0\\ \vdots\\ \sqrt{(x+\frac{b}{2a})^2}=\sqrt{\frac{b^2-4ac}{4a^2}}\\ ...
0
votes
3answers
72 views

Show that $f(x)=2x^2+4x+5$ is positive for all real values of $x$.

Show that the function $f(x)=2x^2+4x+5$ is positive for all real values of $x$. At first I used completing the square technique, ${ax^2 + bx + c}$ is converted to ${a(x + h)^2 + k}$ $2x^2+4x+5$ ...
1
vote
2answers
35 views

Average $y$ from a range of $x$ in a parabola

Given a parabolic/quadratic formula such as $ax^2 + bx + c =y$, how do I get the average value of $y$ given a range of $x$ ($x_{min}$ to $x_{max}$). Real world example: if my formula represents the ...