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

learn more… | top users | synonyms

2
votes
2answers
79 views

Pythagoras numbers and fermats last theorem

I am reading "What Is Mathematics? An Elementary Approach to Ideas and Methods" And I am stuck here, I don't get it. I have posted a screen shot underlining what my doubt is.. I dont get it when the ...
2
votes
3answers
103 views

If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist?

If $a$ and $b$ are the zeroes of $x^2 + ax + b = 0$, then how many pairs of $(a,b)$ exist? One Two Three Infinitely many Also, what are these pairs?
0
votes
2answers
87 views

Proving the second root of a quadratic equation

If $\alpha$ is a root of the equation $4x^2+2x-1=0$, then prove that $4\alpha^3-3\alpha$ is the other root. How do I proceed? The sum of the roots, the product of the roots lead me nowhere. Should I ...
-1
votes
1answer
365 views

Symmetric System of Equations

I'm new on studying Systems of equations. I just want to know the number of real solutions of this system of equations: \begin{align*} x^2-y^2=z\\ y^2-z^2=x\\ z^2-x^2=y \end{align*} I also want to ...
-5
votes
4answers
164 views

Roots of $x^2+3x+2=0$ are infinite !!! [closed]

I have quadratic equation here: $x^2+3x+2=0$ so $(x+2)(x+1)=0$ and I can do $(x+2)=0/(x+1)$ and that solution of the equation is $x+2=0$ so $x=-2$ but my teacher said that it is wrong why? ...
0
votes
3answers
42 views

Find the integer n

Let $a$ and $b$ be two integers such that $10a+b=5$ and $p(x)=x^2+ax+b$. Find the integer $n$ such that $p(10)p(11)=p(n).$ Please tell how to proceed.
1
vote
1answer
52 views

Proving the minimum value of (x+a)(x+b)/(x+c)

Show that the minimum value of $\frac {(x+a)(x+b)}{(x+c)}$, where a$\gt$c, b$\gt$c, is $(\sqrt{a-c}+\sqrt{b-c})^{2}$ for real values of x$\gt-c$. I did $$\frac {(x+a)(x+b)}{(x+c)}=y$$ and then took ...
3
votes
2answers
69 views

Proper method for solving quadratic equations with exponents

$(\sqrt {x^2-5x+6}+\sqrt{x^2-5x+4})^{x/2}$ + $(\sqrt {x^2-5x+6}-\sqrt{x^2-5x+4})^{x/2}$ = $2^{(x+4)/4}$ I have found out, by trial and error method, that $x=0$ and $x=4$ satisfy this equation. But is ...
2
votes
3answers
102 views

Prove that for real numbers $x$, if $x^2 - 5x + 4 \ge 0$, then either $x \le 1$ or $x \ge 4$.

Its another homework question that I'm having trouble understanding. The full question is write a detailed structured proof that uses a proof by cases to prove that for real numbers $x$, if $x^2 - 5x ...
2
votes
2answers
56 views

Quadratic inequalities

This is what I tried. I tried finding limits of y and then equating them with the given limits, but I could not simplify it further. The given options for this question are: a+b=23 a^2+b^2=277 ...
2
votes
3answers
89 views

Quadratic equation problems

There is a circle with a radius of $25$ ft and origin at $(0, 0)$ and a line segment from (0, -31) to (-37, 8). Find the intersections of the line and circle. I am asking for somebody to analyze ...
1
vote
1answer
256 views

Probabilistic Robotics Exercise

I am reading Probabilistic Robotics and I don't know how to solve the exercise problem number 4 at the end of the second chapter. There are no solutions to this text. The exercise states: In ...
6
votes
2answers
92 views

Why/when did these extraneous solutions appear while solving a quadratic equation?

I am trying to solve the quadratic equation $x^2 + x + 1 = 0$. $x^2 = -1 - x $ $\iff x = -\frac{1}{x} - 1$, assuming $x\neq 0$. Substituting that into the original equation gives $x^2 + (-\frac{1}{x} ...
0
votes
1answer
24 views

Find a if the equation has a solution

If this equation has a solution, then 'a' is equal to None of these How should I proceed in this problem?
1
vote
2answers
36 views

Least Value of the Quadratic Expression

What is the least value of this expression? Please show me a way to determine it.
5
votes
2answers
353 views

Factoring Quadratics: Asterisk Method

I'm teaching my students about factoring quadratics. We've done GCF, difference of two squares, squared binomials, and grouping. One of my colleagues then found this asterisk method on line. It's ...
4
votes
2answers
174 views

Solving system of multivariable 2nd-degree polynomials

How would you go about solving a problem such as: \begin{matrix} { x }^{ 2 }+3xy-9=0 \quad(1)\\ 2{ y }^{ 2 }-4xy+5=0 \quad(2) \end{matrix} where $(x,y)\in\mathbb{C}^{2}$. More generally, how would ...
1
vote
4answers
529 views

Real world examples of quadratic and/or finding roots of a quadratic?

Anyone ever come across a good situation where a) a situation is modeled by a quadratic equation $y=ax^2+bx+c$ and/or b) you've even needed to find where $y=0$ (roots, $x$-intercept, etc)
3
votes
1answer
345 views

by completing the square find in terms of k the roots of the equation $x^2 + 2kx-7=0$

By completing the square find in terms of $k$ the roots of the equation $$x^2 + 2kx-7=0$$ prove for all real values of $k$, the roots are real
5
votes
4answers
275 views

Derivation of the quadratic equation

So everyone knows that when $ax^2+bx+c=0$,$$x = \dfrac{-b\pm\sqrt{b^2-4ac}}{2a}.$$ But why does it equal this? I learned this in maths not 2 weeks ago and it makes no sense to me
1
vote
1answer
95 views

Why does completing the square give you the minimum point?

Say we have an equation:$y=$ ${x^2} + 2x + 1$ Completing the square we get: $\eqalign{ & y={x^2} + 2x + 1 \cr & = {(x + 1)^2} - {(1)^2} + 1 \cr & = {(x + 1)^2} \cr} $ The ...
0
votes
1answer
40 views

Algebra steps breakdown

I have been reviewing this question. : How can I find the points at which two circles intersect? Although during the answering steps I got a little stuck: From this: 17y2−62y+49=0 To this: ...
5
votes
1answer
293 views

Finding integral roots of $x^2 + px + q = 0$ if $p+q=198$.

Given the relation that $p+q=198$, the question is to find all the integral roots of the equation: $$ x^2+px + q = 0 $$ How to proceed? I know we'll have to use Vieta's formulas, but I don't know ...
1
vote
1answer
59 views

$\frac{\text{Quadratic}}{\text{Quadratic}}$ methods

So, lets take up a question. Show that $f(x)=\frac{x^2+34x-71}{x^2+2x-7}$ can never lie between 5 and 9. MY ATTEMPT: I assumed the function to be equal to k , then cross multiplied and got a single ...
0
votes
1answer
31 views

One way to describe the pattern of covariation for a linear function is:

One way to describe the pattern of covariation for a linear function is: As input value increases by 1, the output value changes by a constant (fixed) amount k where k is some real number. Explain why ...
1
vote
2answers
1k views

Finding vertex and focus of parabola given an equation

I am defeated to complete square on the following parabolic equation. Please help. Find the vertex and focal width for the parabola: $$ x^2+6x+8y+1=0 $$ I am hoping to get an equation in this form ...
1
vote
1answer
126 views

Find conditions on a,b,c so that p(x) and q(x) have exactly 2 roots in common. Also solve the equation p(x)=0

Let $p(x) = x^4 +ax^3 +bx^2+cx +1$ and $q(x) = x^4 +cx^3 +bx^2+ax+1$ with a,b,c real numbers.Find conditions on a,b,c so that p(x) and q(x) have exactly 2 roots in common. Also solve the equation ...
1
vote
1answer
44 views

Solving Quadratics

A rectangular lawn measures 30 m by 40 m. Jason is cutting the lawn from the outside perimeter in toward the centre by cutting strips along the entire perimeter first, then continuing as he cuts ...
2
votes
2answers
69 views

Solving $y^2(x^2+1) +x^2(y^2+16) =448$

$y^2(x^2+1) +x^2(y^2+16) =448$ The task is to find all solutions in integers $(x,y)$. This is the fourth question of rmo 1st stage.The solution here is not complete. I have tried to solve unable to. ...
0
votes
0answers
35 views

If $ax^2-bx+c=0$ has two distinct roots in the interval (0,1) $a,b,c \space \epsilon \space N$ then the least value of abc = [duplicate]

If $ax^2-bx+c=0$ has two distinct roots in the interval (0,1) $a,b,c \space \epsilon \space N$ then the least value of abc = ? Status : Stuck for a while. No clue. Any help is highly appreciated . ...
0
votes
1answer
70 views

Solving a quadratic made from the sum of monomial denominators. [closed]

Solve the following equation. Separate your answers with commas. Repeated roots should only be entered once. $$\frac{1}{x-5} + \frac{1}{x-6} = \frac{11}{30}$$ Any ideas on how to start out?
3
votes
1answer
89 views

sub rectangle region from combined area

I have a rectangle divided like so ...
-1
votes
2answers
46 views

Confusion with qudratic equations

According to my book In the given equation $$x^2+x+1=0\tag{1}$$ If $a$ is a root of eqn $(1)$ then $a$ satisfies the following equation $$a^2+a+1=0\tag{2}$$ $$\implies ...
1
vote
2answers
832 views

coefficients of quadratic function?

In a quadratic function: coefficient $a$ controls the speed of increase/decrease from the vertex. coefficient $b$ controls the downward slope as the function crosses the y-axis. I don't really ...
2
votes
1answer
184 views

If I have an x intercept and a y intercept, how might I find the vertex of a parabola?

I have looked all over, and I have found different things here and there for figuring pretty much everything but that. What I have is an x chord and a y chord, but I need to find the vertex. This ...
2
votes
1answer
96 views

Graphing quadratic form, which eigenvalue should be chosen first?

I just graph a quadratic function, $-4x^2_1+4x_1x_2-7x_2^2=-8$, by: Find the eigenvalues of the function above, which are $\lambda_1=-8$ and $\lambda_2=-3$ Use the eigenvalues to make a new ...
1
vote
4answers
73 views

Find the roots of the given equation : $2^{x+2}.3^{\frac{3x}{x-1}} =9$ - Logarithm problem

Find the roots of the given equation : $2^{x+2}.3^{\frac{3x}{x-1}} =9$ My working : Taking log on both sides we get : $$\log (2^{x+2}.3^{\frac{3x}{x-1}}) =\log 3^2 \Rightarrow (x+2)(\log2) + ...
3
votes
2answers
1k views

Finding the discriminant and roots of a polynomial

How is the discriminant of a polynomial determined? I know that for a quadratic function, the roots (where $f(x)=0$) are found by $$x=\frac{-b\pm\sqrt{\Delta}}{2a}$$ and here $\Delta$ is the ...
1
vote
2answers
115 views

Can we refer to the standard form of a quadratic equation as the general form as well?

I would like to know if we can refer to $$ax^2+bx+c=0$$ as the "general form" of a quadratic equation, or is it only called the standard form?
1
vote
2answers
59 views

How do you know which substitutions to make to cancel out a term?

I am doing problem B45 from Ivan Niven's "Maxima and Minima Without Calculus" which says: "Consider the quadratic polynomial $f(x, y)=ax^2+2bxy+cy^2+dx+cy+k$ , where the coefficients are real ...
1
vote
2answers
287 views

Solving inequalities, simplifying radicals, and factoring. (Pre calculus)

(Q.1) Solve for $x$ in $x^3 - 5x > 4x^2$ its a question in pre calculus for dummies workbook, chapter 2. The answer says: then factor the quadratic: $x(x-5)(x+1)>0$. Set your factors equal to ...
0
votes
2answers
60 views

Quadratic topic- very basic

Express $x(4-x)$ as the difference of two squares. I do not really quite sure what is meant by difference of two squares.
3
votes
2answers
13k views

Quadratic equation - Alpha and Beta Roots

If α and β are the roots of the equation x² + 8x - 5 = 0, find the quadratic equation whose roots are α/β and β/α. My working out so far: I know that α+β = -8 and αβ = -5 (from the roots) and then i ...
-1
votes
2answers
66 views

I need help with a trig proof. [duplicate]

without a calculator, prove $\sin^2 x- 6\sin x-5=0$ has more than one real solution. I have repeatedly solved this but I have only got one solution. Can someone help me out! Show your work how you ...
2
votes
2answers
112 views

Finesse vs. brute force in solving quadratic equations

In Higher Algebra by Hall and Knight, the following "artifice" for solving a certain type of equations is given: Solve: $\sqrt{3x^2-4x+34} - \sqrt{3x^2-4x-11} = 9$ They make use of the fact ...
-1
votes
1answer
80 views

If $a,b,c \in R$ such that $c \neq0$ If $x_1$ is a root of $a^2x^2+bx+c=0, x_2$ is a root of $a^2x^2-bx-c=0 $ and $x_1 > x_2 >0$…

Problem : If $a,b,c \in R$ such that $c \neq0$ If $x_1$ is a root of $a^2x^2+bx+c=0, x_2$ is a root of $a^2x^2-bx-c=0 $ and $x_1 > x_2 >0$ then the equation $a^2x^2+2bx+2c=0$ has roots $x_3$ ...
2
votes
2answers
56 views

Factor this quadratic expression

I need to do the following: Prove that a quadratic expression of the form $A(x^2-y^2) - (B-C)xy$ can be always factored into two linear factors. It is easy enough to compare this with the ...
1
vote
2answers
113 views

System of quadratic equations

How would you solve the following system of equations: $$ x^2 + y = 4 \\ x + y^2 = 10 $$ Thanks very much! I tried defining y in terms of x and then inserting in to the second equation: $$ y = 4 - ...
2
votes
2answers
152 views

If the roots of the quadratic equation $2kx^{2}+(4k-1)x+2k-3=0$ are rational and k is an integer, how many values can k take which are less that 50?

If the roots of the quadratic equation $2kx^{2}+(4k-1)x+2k-3=0$ are rational and k is an integer, how many values can k take which are less that 50 ? The discriminant = $16k+1$ For a rational number ...
1
vote
3answers
159 views

Solve the equation : $x^2 − 6 |x − 2| − 28 = 0$

The following is an absolute value quadratic equation that I want to solve: $$x^2 − 6 |x − 2| − 28 = 0$$ Here is what I did : $x^2 − 6 |x − 2| − 28 = 0$ $x^2 − 6 |x − 2| − 28 = 0$ ...