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

96
votes
14answers
8k views

Why can ALL quadratic equations be solved by the quadratic formula?

In algebra, all quadratic problems can be solved by using the quadratic formula. I read a couple of books, and they told me only HOW and WHEN to use this formula, but they don't tell me WHY I can use ...
3
votes
5answers
295 views

How to “Re-write completing the square”: $x^2+x+1$

The exercise asks to "Re-write completing the square": $$x^2+x+1$$ The answer is: $$(x+\frac{1}{2})^2+\frac{3}{4}$$ I don't even understand what it means with "Re-write completing the square".. ...
16
votes
3answers
386 views

What would be the value of $\sum\limits_{n=0}^\infty \frac{1}{an^2+bn+c}$

I would like to evaluate the sum $$\sum_{n=0}^\infty \frac{1}{an^2+bn+c}$$ Here is my attempt: Letting $$f(z)=\frac{1}{az^2+bz+c}$$ The poles of $f(z)$ are located at $$z_0 = ...
3
votes
5answers
212 views

Proving Quadratic Formula

purplemath.com explains the quadratic formula. I don't understand the third row in the "Derive the Quadratic Formula by solving $ax^2 + bx + c = 0$." section. How does $\dfrac{b}{2a}$ become ...
5
votes
2answers
294 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
1answer
488 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 ...
2
votes
4answers
139 views

Solving a quadratic Inequality

My question is: Solve $$9x-14-x^2>0$$ My answer is: $2 < x < 7$ Though I know my answer is right, I want to know in what ways I can solve it and how it can be graphically represented. ...
1
vote
3answers
297 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 ...
0
votes
1answer
30 views

With Trinomial's, can someone explain the purpose of the first, second and third term. In layman terms.

I know that the first term is a quadratic and I suppose that lets us know we are dealing with identifying a curve, and the third term is our constant. I just can't quite put it all together as how ...
0
votes
0answers
130 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 ...
5
votes
4answers
262 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
votes
1answer
79 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 $x_3$ ...
3
votes
5answers
446 views

Condition for a common root in two given quadratic equations

If $a,\;b,\;c$ are in Geometric Progression, then the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root if $\;\displaystyle\frac da,\;\frac eb,\;\frac fc$ are in: Arithmetic Progression ...
3
votes
4answers
218 views

Where is the order of the variables inside the parentheses coming from?

I'm reading Sawyer's Prelude to Mathematics, here: I can't understand what's the meaning and application of "condition" here. Also when he gives the example on the cubic equation, stating that ...
3
votes
2answers
261 views

Integrating $\int_0^\infty \frac{1}{x^2 + 2x + 2} \mathrm{d} x$

I've been trying to integrate this: $$\int_0^\infty \frac{1}{x^2 + 2x + 2} \mathrm{d} x .$$ Unfortunately I haven't found a way so far. I've been trying to factor the denominator in order to end up ...
2
votes
1answer
42 views

How do I transform the equation based on this condition?

If a and b are the roots of the equation $$2x^2-px+7=0$$ Then a-b is a root of ?
0
votes
1answer
93 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 $x<a, |x-a|=-(x-a)$ ...
5
votes
5answers
3k views

Why a quadratic equations always equals zero?

On evaluating quadratic equations, It always equals zero: $$ax^2+bx+c=0$$ Why zero? Is it possible to use other number for another purpose?
5
votes
2answers
838 views

Solution af a system of 2 quadratic equations

I have a system of two quadratic equations with unknowns $x$ and $y$: $$a_{1 1} x y + a_{1 2} x^2 + a_{1 3} y^2 + a_{1 4} x + a_{1 5} y + a_{1 6} = 0,\\ a_{2 1} x y + a_{2 2} x^2 + a_{2 3} y^2 + a_{2 ...
5
votes
2answers
549 views

Find equation of quadratic when given tangents?

I know the equations of 4 lines which are tangents to a quadratic: $y=2x-10$ $y=x-4$ $y=-x-4$ $y=-2x-10$ If I know that all of these equations are tangents, how do I find the equation of the ...
4
votes
2answers
121 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 ...
3
votes
1answer
186 views

Curve through four points — simple algebra??

The motivation for this is Bezier curves. But, if you don't know what these are, you can skip down to the last paragraph, where the problem is described in purely algebraic terms. Suppose I want to ...
2
votes
3answers
95 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
1answer
153 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
2answers
182 views

sum of squares of the roots of equation

The equation is $$x^2-7[x]+5=0.$$ Here $[x]$ the greatest integer less than or equal to $x$. Some other method other than brute forcing. I tried a method of putting $[x]=q$ and $x=q+r$ which gives an ...
1
vote
2answers
176 views

Find the maximum or minimum value of the quadratic function.

Find the maximum or minimum value of the quadratic function by completing the squares. Also, state the value of $x$ at which the function is maximum or minimum. $y=2x^2-4x+7$ $x^2$ has a coefficient ...
1
vote
1answer
63 views

What technique reduces factorable ax^2+bx+c=0 to factorable where a=1

The normal way to factor ax^2+bx+c=0 is to look for t,u,v,w such that: (tx+u)(vx+w) = 0 so that tv=a, uw=c, and uv+wt=b. This can be tricky, since there can be several possibilities for t,u,v,w. ...
1
vote
3answers
192 views

How do I solve a Continued Fraction of solution to quadratic equation?

I know that it is possible to make a CF (continued fraction) for every number that is a solution of a quadratic equation but I don't know how. The number I'd like to write as a CF is: $$\frac{1 - ...