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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2
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1answer
137 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
78 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
2k 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
124 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
66 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
382 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
61 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
18k 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
68 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
143 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
63 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
114 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
172 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
191 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 : $x^2 − 6 |x − 2| − 28 = 0$ $x^2 − 6 |x − 2| − 28 = 0$ ...
2
votes
0answers
119 views

Proving an equality involving binomial coefficients and summations

Question: $$\sum_{k=0}^{n}\left ( -1 \right )^{k}\binom{2n}{k}\binom{2n-k}{2n-2k}=\sum_{2n}^{k=0}\binom{2n}{k}^{2}\left ( \frac{1+\sqrt{5}}{2} \right )^{2n-k}\left ( \frac{1-\sqrt{5}}{2} \right ...
0
votes
1answer
119 views

Determine all the values of the parameter $a$ for which the inequality $3-|x-a|>x^2$ is satisfied by at least one negative $x$.

I wanted to know, how can I determine all the values of the parameter $a$ for which the inequality $3 - |x-a| > x^2$ is satisfied by at least one negative $x$. I tried for $x<a, |x-a|=-(x-a)$ ...
2
votes
1answer
39 views

Help with this solution of quadratic equation

In one of the solved examples in the book I'm following, the following expression arises after considering $D > 0$ for a certain equation: $$D = (n+1)^2p^2 - 2pq(n^2+1) + (n-1)^2q^2$$ From this, ...
2
votes
3answers
216 views

Given that the roots of the quadratic equation $x^2+2ax+3a=0$ lie between $-1$ and $1$, what are the possible values of $a$?

In the equation $x^2+2ax+3a=0$ has two solutions $\alpha$ and $\beta$ where $-1<\alpha,\beta<1$. Find out the range of $a$. I tried to solve it by taking $\alpha^2+\beta^2$. But it does ...
2
votes
2answers
134 views

maximum using completing the square

Is it just me, or this problem does sound weird? The Parks Department is fencing a rectangular dog-run (a place for dogs to exercise) in a local park. It will be 7 yards longer than 5 times its ...
3
votes
4answers
62 views

How to search quadratic function

If a graph of the quadratic function $f(x)=ax^2+bx+c$, where $a$, $b$ and $c$ are constants. If this function vertex is $(13,−169)$ and the distance between the two intersection points with the ...
1
vote
3answers
390 views

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots.

If $(2x^2-3x+1)(2x^2+5x+1)=9x^2$,then prove that the equation has real roots. MY attempt: We can open and get a bi quadratic but that is two difficult to show that it has real roots.THere must be an ...
5
votes
3answers
788 views

How to solve problems involving roots. $\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$

How to solve problems involving roots. If we square them they may go to fourth degree.There must be some technique to solve this. $$\sqrt{(x+3)-4\sqrt{x-1}} + \sqrt{(x+8)-6\sqrt{x-1}} =1$$
0
votes
1answer
49 views

The number of integral values of $a$ for which the inequality $3- |x-a |>x^2$ is satisfied by at least one negative $x$, must be equal to 6

The number of integral values of $a$ for which the inequality $3- |x-a |>x^2$ is satisfied by at least one negative $x$, must be equal to 6. I don't know how to solve this. Can you help?
4
votes
1answer
634 views

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ $a,b,c$ belongs to natural prove that $\log_5 {abc}\geq2$

If $ax^2-bx+c=0$ has two distinct real roots lying in the interval $(0,1)$ with $a, b, c\in \mathbb N$, prove that $\log_5 {abc}\geq2$. The equations I could form are: 1) $f(0)>0$ and ...
1
vote
2answers
75 views

The no. of values of k for which $(16x^2+12x+39) + k(9x^2 -2x +11)$ is perfect square is:

I wanted to know, how can i determine the no. of values of k for which $(16x^2+12x+39) + k(9x^2 -2x +11)$ is a perfect square.($x \in R$) I have tried, since $x$ is real the discriminant must be ...
4
votes
2answers
186 views

Find all values of a for which the equation $x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0$ possesses at least two distinct negative roots

Find all values of a for which the equation $$x^4 +(a-1)x^3 +x^2 +(a-1)x+1=0 $$ possesses at least two distinct negative roots. I am able to prove that all roots would be negative .How to proceed ...
0
votes
0answers
302 views

If $ a+b+c=4 ; a^2=b^2+c=6 ; a^3+b^3+c^3=8 $ Then find the value of $a^4+b^4+c^4$

If $a+b+c=4$ $ a^2=b^2+c=6$ (this is not symmetric equation) $a^3+b^3+c^3=8$ Then find the value of $a^4+b^4+c^4$
3
votes
2answers
130 views

For what $x\in[0,2\pi]$ is $\sin x < \cos 2x$

What's the set of all solutions to the inequality $\sin x < \cos 2x$ for $x \in [0, 2\pi]$? I know the answer is $[0, \frac{\pi}{6}) \cup (\frac{5\pi}{6}, \frac{3\pi}{2}) \cup (\frac{3\pi}{2}, ...
1
vote
1answer
75 views

The least value of $4x^2-4ax +a^2-2a+2$ on $[0,2]$ is $3$. What is the integer part of $a$?

The least value of $4x^2-4ax +a^2-2a+2$ on $[0,2]$ is $3$. What is the integer part of $a$? We know that minimum value of a quadratic is $-\cfrac{b}{2a}$. We will get one condition from here and ...
0
votes
1answer
115 views

Quadratic Polynomial Question - Solving for a coefficient using the discriminant

This question has been troubling me: A parabola whose equation is of the form $y = Bx^2$ (where B is a constant) has the line $20x - y + 20 = 0$ as a tangent. Find $B$. The explanation says, ...
0
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1answer
224 views

Quadratic expression that generate primes

I recently learned that there exist quadratic expression that generate some primes and some of these equations generate more primes than others. In the following video, the person shows the following ...
2
votes
4answers
210 views

$a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0 $

Let $a,\,b,\,c,\,d$ be distinct real numbers and $a$ and $b$ are the roots of quadratic equation $x^2 -2cx-5d=0$ and $c$ and $d$ are the roots of quadratic equation $x^2 -2ax-5b=0$. Then find the ...
1
vote
4answers
204 views

Find all real numbers such that $\sqrt {x-1/x } + \sqrt {1 - 1/x} = x$

Find all real numbers such that $$\sqrt{x-1/x} + \sqrt{1 - 1/x} = x$$ My attempt to the solution : I tried to square both sides and tried to remove the root but the equation became of 6th ...
3
votes
1answer
243 views

$y =f(x) =(ax^2 + bx +c)/(dx^2+ex+f)$ We have to find the conditions for this it takes all real values.

$$ y=f(x)=\frac{ax^2+bx+c}{dx^2+ex+f} $$ We have to find the conditions for this it takes all real values. MY solution One approach is to equate it to y and for a quadratic of x and put discriminant ...
0
votes
1answer
58 views

Find the integer solutions

What are the pairs $(A,N)$ where $A,N$ are integers such that the following equation is satisfied: $\large A=\frac{-6+\sqrt{144-12N^2}}{6}$ I know that we should have: $k^2=144-12N^2$ for some ...
1
vote
1answer
277 views

Show that that if $p,q,r,s$ are real numbers and $pr=2(q+s)$, then at least one of the eqns $x^2+px+q=0$ and $x^2+rx+s=0$ has real roots.

Show that that if $p,q,r,s$ are real numbers and $pr=2(q+s)$, then at least one of the eqns $x^2+px+q=0$ and $x^2+rx+s=0$ has real roots. My Attempt to the solution we know to have a real solution ...
1
vote
2answers
106 views

How many real roots for $ax^2 + 12x + c = 0$?

If $a$ and $c$ are integers and $2 < a < 8$ and $-1 < c$, how many equations of the form $$ax^2+12x+c=0$$ have real roots?
3
votes
3answers
186 views

Systems of Quadratic Equations Question

looking for help on this question. Solve the following systems of equations algebraically using the quadratic formula. $$\begin{align} y& =-x^2+2x+9\\ y& =-5x^2+10x+12\end{align}$$ Any help ...
0
votes
1answer
325 views

Finding the Equation of Parabola

Write the equation in the form $y=a(x-h)^{2}+k$ with zeros -4 and 8, and an optimal value of 18. I'm not sure what "optimal value" means first of all- I think it means that the maximum value has ...
0
votes
3answers
78 views

How to isolate $v_m$?

Note: I am not asking anything pertaining to the physics of this question; only the mathematics. The physics is just given as a context to the problem for those interested, as opposed to simply saying ...
5
votes
3answers
126 views

How do you solve $4x^2=-16x$? I get different answers depending on the method used.

I'm solving the following GRE problem: Solve $4x^2=-16x$ Method 1: I simply divide both sides by $4x$ :$$x=-4$$ Method 2: I solve by factoring:$$4x^2+16x=0$$ $$4x(x+4)=0$$ $$x=-4, x=0$$ Using ...
1
vote
3answers
168 views

Quadratic Equation find the value of $\lambda$ when other roots are given in restriction

Problem : If $\lambda$ be an integer and $\alpha, \beta$ be the roots of $4x^2-16x+\lambda$=0 such that $ 1 < \alpha <2$ and $2 < \beta <3$, then find the possible values of $\lambda$ ...
0
votes
0answers
164 views

Sample Code to Generate Points on the Rim of a Randomly Rotated Cone : What's Going On Here?

Related to this question: http://math.stackexchange.com/questions/407897/randomly-generate-point-on-shell-from-3-points-2-angles-with-uniform-angle-dis I'm trying to reverse engineer the ...
1
vote
2answers
72 views

Quadratic expression in $x$ with roots $\frac gh$ and $-\frac hg$

What quadratic expression in $x$ has roots $$\frac{g}{h}\qquad\text{and}\qquad-\frac{h}{g}?$$ I know that this can be factored as $$\left ( x-\frac{g}{h} \right )\left ( x+\frac{h}{g} \right )=0$$ But ...
5
votes
2answers
86 views

System of Pythagorean Quadratics

I have a system of quadratics, obtained from three mechanical links, fixed at one end and free at the other. The intersection point of the three free ends is required. ...
1
vote
1answer
235 views

Finding the rational values of constant for which these constants are roots of equation

Problem : Determine all rational values for which $a,b,c$ are the roots of $x^3+ax^2+bx+c=0$ Solution : Sum of the roots $a+b+c = -a$ ........(i) ( Since , as per question $a,b,c$ are roots of ...
0
votes
2answers
38 views

System of equations in x and y

Solve for $x, y \in \mathbb{R} $ $$ 5x \left(1+\frac{1}{x^2+y^2}\right) =12$$ $$ 5y \left(1-\frac{1}{x^2+y^2}\right) =4$$ I need a Different Approach apart from what i posted..Thank You
3
votes
3answers
235 views

Is it possible to find out $x^2$ parabola and function from 3 given points?

I am programming a ball falling down from a cliff and bouncing back. The physics can be ignored and I want to use a simple $y = ax^2$ parabola to draw the falling ball. I have given two points, the ...
1
vote
4answers
67 views

Quadratic Formula problem?

There is a right triangle. The hypotenuse is 17 units. The sum of the other two sides is 23. Find the length of the two other sides. Thanks for everyone's help in advance!