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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114
votes
16answers
11k 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 ...
5
votes
2answers
380 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 ...
11
votes
8answers
597 views

How to factor quadratic $ax^2+bx+c$?

How do I shorten this? How do I have to think? $$ x^2 + x - 2$$ The answer is $$(x+2)(x-1)$$ I don't know how to get to the answer systematically. Could someone explain? Does anyone have a link to ...
3
votes
5answers
300 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".. ...
17
votes
3answers
399 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
230 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 ...
3
votes
5answers
797 views

Where did $-4x$ come from?

I'm going over my quadratic equations for the ACT and I came across this quadratic: $$(x – 2)^2 – 12$$ My teacher said we could have factored it out into this: $$x^2 – 4x – 8$$ But I just don't ...
4
votes
1answer
585 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 ...
5
votes
2answers
156 views

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$.

Let $f(x)=ax^2+bx+c$ where $a,b,c$ are real numbers. Suppose $f(-1),f(0),f(1) \in [-1,1]$. Prove that $|f(x)|\le \frac{3}{2}$ for all $x \in [-1,1]$. I made quite a few attempts but could not ...
2
votes
2answers
79 views

Find the value of $f(x)$ for $x = 2 + 2^{2/3} + 2^{1/3}$

If $x = 2 + 2^{2/3} + 2^{1/3}$, then find the value of $f(x)=x^3 - 6x^2 + 6x$. I am unable to get to the answer - end up with more than one term. Please help me solve this!
2
votes
4answers
200 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
368 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
5answers
72 views

How do you factor a quadratic expression, without using the formula?

I am asked to factor $2x^2 -3x+1=0 $ using factorization, but I run into fractions, and it becomes very messy and complicated to deal with, especially since specifically asked not to use the formula. ...
0
votes
1answer
32 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
155 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
votes
2answers
51 views

Proof of axis of symmetry equation [closed]

Because quadratic functions are symmetrical how do you prove the axis of symmetry equation. $x=(-b/(2a))$
-2
votes
2answers
57 views

Using the quadratic equation to find the time for the ball to drop to a fifth of the height of the building

A ball is thrown down at 72km h-1 speed from the top of a building. The building is 125 metres tall. The distance travelled before it reached the ground is as follows... $$s = U_0 t + ...
5
votes
4answers
276 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
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 $x_3$ ...
3
votes
5answers
1k 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
246 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
262 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
2answers
239 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 ...
2
votes
1answer
44 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 ?
2
votes
6answers
271 views

Factoring Quadratics

Is there a method to find which numbers to use when simplifying quadratics? For example $x^2 + 5x + 6$ is easy enough to find, but what if I have bigger numbers, or I have this quadratic expression: ...
1
vote
5answers
2k views

On the number of possible solutions for a quadratic equation.

Solving a quadratic equation will yield two roots: $$\frac{-\sqrt{b^2-4 a c}-b} {2 a}$$ and: $$\frac{\sqrt{b^2-4a c}-b}{2 a}$$ And I've been taught to answer it like: $$\frac{\pm\sqrt{b^2-4a c}- ...
0
votes
1answer
105 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
4k 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
900 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
608 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
188 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
2answers
3k 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 ...
3
votes
1answer
214 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 ...
3
votes
3answers
135 views

Solving quadratic equations by completing the square.

Graphing $y=ax^2+ bx + c$ by completing the square Add and subtract the square of half the coefficent of $x$. Group the perfect square trinomial. Write the trinomial as a square of a ...
2
votes
1answer
65 views

Calculate the volume of water in glass over time.

For A) I found that volume should be defined by But I got no idea what to do in b) and c)
2
votes
4answers
52 views

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take?

If $9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$, then what values can $t$ take? This is what I have done: Let $y = 3^x$ $$9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$$ $$\implies9y^2 + (t^2 - 4t - 2)y + ...
2
votes
3answers
104 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
190 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 ...
1
vote
1answer
90 views

combination of quadratic and cubic series

I'm an eight-grader and I need help to answer this math problem (homework). Problem: Calculate $$\frac{1^2+2^2+3^2+4^2+...+1000^2}{1^3+2^3+3^3+4^3+...+1000^3}$$ Attempt: I know how to calculate ...
1
vote
4answers
145 views

How can I solve equation $x^2 - y^2 -2xy - x + y = 0$?

I have this equation with 2 variables - $$x^2 - y^2 -2xy - x + y = 0$$ The only condition I have is that $x + y$ should be greater than $10^{12}$. EDIT - I need $x$ and $y$ to be integer. I ...
1
vote
1answer
66 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
210 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 - ...
0
votes
3answers
77 views

Absolute value quadratic inequalities not the usual?

$ | -x^2 + 6x | \gt 13 $,for example. I would start off solving $ -x^2 + 6x = \pm 13 $ and either get 4 solutions, 3 solutions or two simply do the the nature of the graph. Without knowing if the two ...
0
votes
0answers
49 views

A question using quadratic equations. [duplicate]

A ball is thrown down at 72km h-1 speed from the top of a building. The building is 125 metres tall. the distance travelled before it reached the ground is as follows... s = Uot + 1/5gt2 where Uo ...