Mathematics Stack Exchange
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Questions tagged [quadratics]

Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

Questions about quadratic functions and equations, second degree polynomials usually in the forms $y=ax^2+bx+c$, $y=a(x-b)^2+c$ or $y=a(x+b)(x+c)$.

The root of $y=ax^2+bx+c$ can be solved by the formula $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

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Why do we say that in a perfect squared quadratic, for example $(x+5)^2$ has two roots when it clearly crosses the x axis once?

Consider the following function: $$f(x) = (x+5)^2$$ Its graph will be a parabola only crossing the $x$ $axis$ once. By setting the equation to $0$ and using our good old $quadratic$ $formula$ gives the same roots repeated. But why is this…
Sanji Vinsmoke
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Is there a valid, "shortcut" method to convert a quadratic from vertex form to factored form?

Recently, I had a couple of students give me an interesting way of converting from vertex form to factored form that I've never seen before. One told me he got it from his science teacher, and another one got it from his dad. Neither of them could…
kitukwfyer
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Quadratic formula -rocket height at given equation

A rocket's height is $190$ m that is defined by an equation $c^2 + 160c +20$. What will the equation be when the rocket's height is $60$ m? Can anyone give me some clue in solving the above problem?
user7519
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What is the Logic to be used for Solving this Problem?

I came across the following in a quiz contest qualification test: $$x = 2 + {1\over 2+ {\cfrac{1}{2+\cfrac{1}{2+\cfrac{1}{\ddots}}}}}$$ Find the value of: $$\frac{3x^2+5x -3}{2x^2 -4x+5}$$ Now, I know that solving the equations is not as important…
Graviton
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Quadratic Equation - I dont know what I'm doing wrong.

So I am learning Quadratic Equations and I have learned about the Formulas for calculating the Delta $(\Delta)$ and $x_1$, $x_2$. I have this equation $$x^2 - 10x + 15 = 0$$ and I've tried to do my best but it turns out that I have the wrong result…
Feelsbadman
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A right triangle has leg $n$, hypotenuse $2n+6$, and perimeter $60$. Find the lengths of all sides.

This problem comes from a Grade 11 math textbook in a section on solving quadratic functions by factoring. What follows is my attempt at a solution. A right triangle has one leg, $n$ and a hypotenuse, $2n+6$. The challenge is to determine the…
MathAdam
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Why are quadratics factored into 2 brackets?

Why is it that no one seems to factor quadratics into just one bracket Eg: $$2x^2+8x+6$$ into $$2x\left(x+4+3\cdot\frac{1}{x}\right)\quad\text{or}\quad 2x\left(x+4+\frac{3}{x}\right)\quad ?$$
Jonathan.
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How to find a when the quadratic equation has equal roots?

Let $a$ be a constant. If quadratic equation $(ax-1)^2+a^2 -a-2 = 0$ has equal roots, then $a=$?
king yau
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"Citardauq" formula derivation?

I'm trying to understand how we got to the "citardauq" formula   (note: "quadratic", reversed) I found this question here, first answer by Andre says Multiply "top" and "bottom" by $-b\mp\sqrt{b^2-4ac}$. After the smoke clears, we obtain…
vexe
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Condition for exactly one root being common b/w two quadratic equations

Question Statement:- If the equations $ax^2+2bx+c=0$ and $a_1x^2+2b_1x+c_1=0$ have one and only one root common, then prove that $b^2-ac$ and $b_1^2-a_1c_1$ are perfect squares. Attempt at a solution:- Let the common root be $\alpha$. Then…
user350331
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Solve a system of quadratic equations in two variables for integral solutions.

Let's say I have two equations like: $x^2 - y^2 = 6$ or $x^2 + y^2 = 6$ What is the best way to solve these sort of equations for finding only positive integral solutions?
Akshay Bhasin
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Limit of the root of quadratic equation

The root of the equation $ a x^2 + bx + c = 0 $ is given by $$ x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \;\;\;...(1) $$ On the other hand, if $a = 0$, then from the original equation we get $$ x = - \frac{c}{b} \;\;\;...(2) $$ So I am guessing that as…
Kota Mori
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When are we able to find a quadratic with roots that are a function of another quadratic?

Motivation: Given the roots of the quadratic $2x^2+6x+7=0$ find a quadratic with roots $\alpha^2-1$ and $\beta^2-1$ I was able to solve this problem in two ways: Method 1: Sum of the roots $\alpha+\beta=-\frac{b}{a}$ Product of roots…
Karl
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Quadratic equation too hard

I am trying to solve a quadratic equation very hard. Is there any other way to solve this without quadratic formula? $$ x^2(-BE(F+C)^2(G+C)(A+C))))+x(C(F+C))\left [ EBA(D-H)-(G+C)(A+C)(B(D-H)+D(F+C)) \right ]+(ABC^2(D-H))=0 $$ A, B, C, D, E, F, G, H…
pipita
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Let $\alpha$ and $\beta$ be the roots of a quadratic equation $4x^2-(5p+1)x+5=0$.If $\beta=1+\alpha,$then find the integral value of $p.$

Let $\alpha$ and $\beta$ be the roots of a quadratic equation $4x^2-(5p+1)x+5=0$.If $\beta=1+\alpha,$then find the integral value of $p.$ Sum of roots$=\alpha+\beta=\frac{5p+1}{4}$ Given $\beta=1+\alpha,$so $\alpha-\beta=-1$ Adding the two…
Vinod Kumar Punia
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