Mathematics Stack Exchange
Mathematics Stack Exchange

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 $$x = \frac{-b\pm \sqrt{b^2-4ac}}{2a}$$

5400 questions
0
votes
3 answers

Let $(x_0,y_0)$ be the solution of the following equations $(2x)^{\ln 2}=(3y)^{\ln 3}, 3^{\ln x}=2^{\ln y}$. Then $x_0$ is:-

Let $(x_0,y_0)$ be the solution of the following equations $(2x)^{\ln 2}=(3y)^{\ln 3}, 3^{\ln x}=2^{\ln y}$. Then $x_0$ is:- My attempt is as follows:- $$x^{\ln3}=y^{\ln2}$$ $$y=x^{\frac{\ln3}{\ln2}}$$ $$(2x)^{\ln 2}=\left(3x^{\frac{\ln 3}{\ln…
user3290550
  • 3,452
0
votes
1 answer

Equation $3\sqrt{-2(x+3)}-1=|x+3|+a$ has exactly two real roots, then the maximum possible value of $|[a]|$ is?

Equation $3\sqrt{-2(x+3)}-1=|x+3|+a$ has exactly two real roots, then the maximum possible value of $|[a]| $ is? ( where [] denotes the greatest integer function ) My attempt is as follows:- $$-(x+3)>=0,…
prat
  • 115
0
votes
1 answer

unsure of reasoning behind a "shortcut" method to solve $(15y+24)^2 = 12(20y^2 +64y +51)$

We are able to solve this problem by expanding, however, I've seen a shortcut method but I'm not sure how this is done. (perhaps comparing coefficients) The shortcut method used is: $(15+24)^2=12∗51−24^2$ <- how does this…
MPP
  • 75
0
votes
2 answers

Factor quadratic equation formula

$ax^2+bx+c=(dx+e)(fx+g)$, and my goal is to find a formula to find $d,e,f,$ and $g$ when given $a,b,$ and $c$ and that also works for complex numbers
Ekadh Singh - Reinstate Monica
  • 640
0
votes
1 answer

What are the limitations of applied quadratics?

I am doing an maths investigation where I use applied quadratics to find the maximum area that a paddock can be when the perimeter must add to $100$ metres. Using the $A = L \times W$ formula, and writing length in terms of width, a quadratic…
Breno J
  • 1
0
votes
3 answers

Trying to analyze degenerate case of quadratic equation $ax^2+bx+c=0$ when $a=0,b=0,c \ne 0$

Trying to analyze degenerate case of quadratic equation $ax^2+bx+c=0$ when $a=0,b=0,c\ne 0$ My attempt is as follows:- $$x=\dfrac{-b\pm\sqrt{b^2-4ac}}{2a}$$ Let's take first root $$x=\lim_{a\to 0,b\to…
user3290550
  • 3,452
0
votes
2 answers

Let f(x) be a quadratic function and c constant. Which of the following statements is correct? (Statements in body of question)

a) There is a unique value of c such that y = f(x) - c has a double root. b) There is unique value of c such that y = f(x-c) has a double root. However, I believe that both of the statements are correct. I considered when f(x) = x^2 and c = 0, and…
Flinn Bella
  • 359
0
votes
1 answer

I want to make an equation from ordered pairs

for a project I'm working on I need to make a equation that passes through 15 different points on a graph. To keep it simple if someone can show me how to do it with 4 ordered pairs I can probably figure out the rest. Thank you!!!
Pie Pie
  • 29
0
votes
1 answer

If $a,b,c$ are in Geometric Progression, then prove that the equations $ax^2+2bx+c=0$ and $dx^2+2ex+f=0$ have a common root

if $\frac da, \frac eb,\frac fc$ are in an Arithmetic Progression $$b=\sqrt {ac}$$ $$ax^2+2\sqrt {ac} x +c=0$$ $$(\sqrt a x+\sqrt c)(\sqrt ax+\sqrt c)=0$$ $$x=-\sqrt{\frac ca}$$ since both roots are roots are equal, both equations have the same…
Aditya
  • 6,191
0
votes
1 answer

Need help with this quadratic questions. Many thanks.

A concert is soon holding and the venue can hold 12000 people. The minimum price is 100 and all tickets can be sold out if the price of each ticket is set to this minimum. For every increment of 1 in the price, the number of tickets sold decreases…
catsssss
  • 1
0
votes
2 answers

mathematical adjustment (IRR)

Is here anybody who can help me with mathematical adjustment between equation 8 and 9 in the picture? Ignore the text between equations
Miroslav Holub
  • 15
0
votes
1 answer

Quadratic equation - LHS RHS cancelling

I am unable to solve this equation: $$\displaystyle\frac{(12x^2+5x-3)}{(4x+3)} = \frac{(6x^2+13x-5)}{(2x+5)}$$ On solving the LHS and RHS become equal and cancel each other. Much grateful if anyone can help me out. Shas
shaswata pal
  • 113
0
votes
4 answers

The difference between the corresponding roots of $x^2+ax+b=0$ and $x^2+bx+ a=0$ is same and $a\not = b$ then prove that $a+b+4=0$

Subtracting both equations $$x(a-b)+b-a=0$$ $$(x-1)(a-b)=0$$ Since $a\not = b$ $$x=1$$ Substitution of x gives $$1+a+b=0$$ which is contradictory to the question. What did I do wrong? There is a solution for this question that proves the required…
Aditya
  • 6,191
0
votes
1 answer

Number of values of x satisfying the pair of quadratic equations $x^2-px+20$ and $x^2-20x+p$ for $p\in R$

Subtracting both equations $$x(20-p)+20-p=0$$ $$x=-1$$ That’s one root, but the answer is three. Where do the the other two roots come from? Edit- I tried using the common roots equation, and I got the values of p as 20 and -21. But only 2 works…
Aditya
  • 6,191
0
votes
1 answer

If $a,b,c, a_1, b_1, c_1$ are rational and equations $ax^2+bx+c=0$ and $a_1x^2+b_1x+c_1=0$ have only one root in common then

prove that $b^2-ac$ and $b_1^2-a_1c_1$ are perfect squares. The only way I could work around this problem was assuming the roots would be rational, which would the discriminat would be a perfect square. Indeed, it is the right way to solve it, but…
Aditya
  • 6,191
1 2 3
…
64 65