# Tagged Questions

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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### 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 ...
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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 ...
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### 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 ...
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### 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".. ...
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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: ...
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### 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. ...
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### How to solve $\ x^2-19\lfloor x\rfloor+88=0$

I have no clue on how to solve this. If you guys have, please show me your solution as well. $$\ x^2-19\lfloor x\rfloor+88=0$$
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### 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
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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 $\dfrac{b^... 5answers 868 views ### Where did$-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 ... 2answers 86 views ### If$\alpha_1,\alpha_2,\ldots,\alpha_n$be the roots of the equation$x^n+1$then$(1-\alpha_1)(1-\alpha_2)\ldots(1-\alpha_n)$equals to ? I think here we need the info of whether$n$is even or odd else how will we say whether by vieta's formula what is the value of$1+(-1)^n$... 1answer 103 views ### Two circles intersection Could you tell what are all the four points in following? Two circles intersect at two points maximum when we want to draw intersecting circles. But there we are solving quadratic equations, what is ... 2answers 303 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 ... 3answers 708 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 ... 1answer 158 views ### If$|ax^2+bx+c|\le 1\ \forall |x|\le 1$, then what is the maximum possible value of$\frac 83a^2+2b^2$? [closed] Let$f(x) = ax^2 + bx + c$;$a,b,c\in\mathbb R$It is given that$|f(x)| \le 1\forall |x| \le 1$Q1) The possible value of$|a+c|$, if$\displaystyle \frac{8}{3} a^2 + 2b^2$is maximum, is ... 2answers 1k views ### Solution af a system of 2 quadratic equations I have a system of two quadratic equations with unknowns$x$and$y$: $$a_{1 1} x y + a_{1 2} x^2 + a_{1 3} y^2 + a_{1 4} x + a_{1 5} y + a_{1 6} = 0,\\ a_{2 1} x y + a_{2 2} x^2 + a_{2 3} y^2 + a_{2 ... 2answers 246 views ### When the quadratic formula has square root of zero, how to proceed? Is there an easier way to solve the following equation?$$x^2=2x-1$$I think I know how to find x, using the quadratic formula: I get$$x^2-2x+1=0$$then$$x=\frac{2 \pm \sqrt{4-4})}2= \frac{... 3answers 269 views ### Proving the quadratic formula (for dummies) [duplicate] I have looked at this question, and also at this one, but I don't understand how the quadratic formula can change from$ax^2+bx+c=0$to$x = \frac{-b\pm\sqrt{b^2-4ac}}{2a}$. I am not particularly good ... 3answers 444 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: $$\frac{1 - \... 3answers 911 views ### EdExcel GCSE question about Hannah and the sweets: show that n^2-n=90 This is my reconstruction of the EdExcel GCSE question that has caused such a Twitter storm in the UK in the last 24 hours, along with its solution. Hannah has a bag containing n sweets, 6 of ... 3answers 70 views ### Rotate the graph of a function? How do I rotate a graph of a function around a point, and show it in the related equation? An example could be f(x)=\lvert x\rvert (absolute Value) and f(x)=x^2 1answer 91 views ### Question about quadratic equation of complex coefficients. Let az^2+bz+c=0 be a quadratic equation with complex coefficients a,b,c and roots z_1, z_2. If it is given that |z_1|\not=|z_2|, how can I obtain the condition for this containing a,b,c? ... 2answers 257 views ### Basis for the Space of Quadratic Polynomials P^{(2)} — Homework Help Prove that 1+t^2, t+t^2, 1+2t+t^2 is a basis for the space of quadratic polynomials P^{(2)}. I have worked it out to the point where I have the following: (1+t^2)(1, 0, 1)^T +(t+t^2)(1,1,0)^... 1answer 43 views ### With Trinomial's, can someone explain the purpose of the first, second and third term. In layman terms. I know that the first term is a quadratic and I suppose that lets us know we are dealing with identifying a curve, and the third term is our constant. I just can't quite put it all together as how ... 12answers 5k views ### Why do I get one extra wrong solution? I'm trying to solve this equation:$$2-x=-\sqrt{x}$$Multiply by (-1):$$\sqrt{x}=x-2$$power of 2:$$x=\left(x-2\right)^2$$then:$$x^2-5x+4=0$$and that means:$$x=1, x=4$$But 1 is not a ... 1answer 381 views ### Nested solutions of a quadratic equation. A quadratic equation of the form x^2+bx+c=0 can be solved with the classical formula that gives all solutions. Here I want discuss some other methods to find one solution. The best known is by ... 2answers 378 views ### Solve x^4+3x^3+6x+4=0… easier way? So I was playing around with solving polynomials last night and realized that I had no idea how to solve a polynomial with no rational roots, such as$$x^4+3x^3+6x+4=0$$Using the rational roots test, ... 3answers 3k views ### How to solve equations to the fourth power? Is it possible to manually retrieve the value of y from the following equation$$\color{blue}{153y^2-y^4=1296}$$WolframAlpha has four solutions for y: -12, -3, 3, 12. How has it solved? What ... 10answers 4k views ### Taking Calculus in a few days and I still don't know how to factorize quadratics Taking Calculus in a few days and I still don't know how to factorize quadratics with a coefficient in front of the 'x' term. I just don't understand any explanation. My teacher gave up and said just ... 4answers 1k views ### Find C such that x^2 - 47x - C = 0 has integer roots, and further conditions Have been working on this for years. Need a system which proves that there exists a number C which has certain properties. I will give a specific example, but am looking for a system which could ... 5answers 160 views ### Finding range of m in x^2+mx+6. Find the range of values of m in the quadratic equation x^2+mx+6=0 such that both the roots of the equation \alpha,\beta<1. My attempt - it is given that \alpha<1 and \beta<1 \... 5answers 10k views ### Why a quadratic equations always equals zero? On evaluating quadratic equations, It always equals zero:$$ax^2+bx+c=0$$Why zero? Is it possible to use other number for another purpose? 2answers 107 views ### How to prove that the roots of this equation are integers? Let there be an equation a^2 + 4ab + b^2 - 121 = 0 where I want to prove that a,b are integers. Then I want to find whether there are integer values of b for which a is also an integer. Let us ... 2answers 79 views ### p can take any value in the interval? If the equation (\cos(p)-1)x^2+\cos(p)x+\sin(p)=0 in the variable x has real roots, then p can take any value in what interval? I applied the discriminant D>0. I get \cos^2( p)-4(\cos(... 3answers 86 views ### How to get the correct angle of the ellipse after approximation I need to get the correct angle of rotation of the ellipses. These ellipses are examples. I have a canonical coefficients of the equation of the five points.$$Ax ^ 2 + Bxy + Cy ^ 2 + Dx + Ey + F = 0$...
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I recently placed a question based on quadratics and received a few valuable answers. One of them was a comment in an answer with a link in it which I found useful. But unfortunately the webpage (of ...
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### Missing solution in quadratic equation

This is a reformulation of this question to better fit this forum. I removed the mentioning of sage math and this question is now 100% math. Given, the following quadrilateral: I want to describe ...
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### Factor $6x^2​ −7x−5=0$

I'm trying to factor $$6x^2​ −7x−5=0$$ but I have no clue about how to do it. I would be able to factor this: $$x^2-14x+40=0$$ $$a+b=-14$$ $$ab=40$$ But $6x^2​ −7x−5=0$ looks like it's not ...
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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 $y = 3^x$ $$9^{x+1} + (t^2 - 4t - 2)3^x + 1 > 0$$ $$\implies9y^2 + (t^2 - 4t - 2)y + 1 &... 3answers 77 views ### For what real values of a does the range of f(x) contains the interval [0,1]? Question : For what real values of a does the range of f(x) = \cfrac{x+1}{a+x^2}  contains the interval [0,1]? My doubt lies in the further preceding of this question. The book states : ... 2answers 299 views ### Integrating \int_0^\infty \frac{1}{x^2 + 2x + 2} \mathrm{d} x I've been trying to integrate this:$$\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 ...
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 ...