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
1 answer

How to check for range condition when a equation containing other function is converted into quadratic?

I came across a question in my textbook which is something like this The least positive value of a for which $4^x-a\times2^x-a+3 \le 0$ is satisfied by at least one real value of $x$ is ______ So to solve this question I used this…
Kumar Shuvam
  • 45
0
votes
1 answer

quadratic formula and simplification

Consider the following equation: $$x=\frac{(p+x(1-p))\cdot c}{1-(p+x(1-p))(1-c)}$$ Solution should be $$x=\frac{cp}{(1-c)(1-p)}$$ I tried to rewrite the first equation in such that:$$x^2[c(1-p )-(1-p)] + x[2cp - c + 1-p]-pc=0$$ Applying quadratic…
Tim
  • 43
0
votes
1 answer

How do you write a quadratic formula with $a, b$, and $c$ being integers and its solutions being rational numbers that are not integers?

This question has me stumped and I feel like I must be missing something obvious. How do you write a quadratic equation in the standard form $ax^2+bx+c=0$ such that $a,b$ and $c$ are integers, but the solutions are rational numbers that are not…
Melissa McPheeters
  • 1
0
votes
2 answers

is it possible to fit 6n(n-1) + 1 = 37 into the quadratic formula?

This question results from my original question here. I'm trying to find x for ... 6x(x-1) + 1 = 37 ... the answer is 3 but I need to know how to get to that answer. I've been advised to use the quadratic formula to do this and to use that I need…
danday74
  • 145
0
votes
2 answers

Prove that quadratic field Q(√2) is closed

For the quadratic field Q(√2), how can I prove that it is closed? This means that any addition, subtraction, multiplication, or division between any a+b√2 and c+d√2 will still be from the set Q(√2)? Addition and subtraction are pretty trivial, so I…
John Liu
  • 419
  • 3
  • 11
0
votes
0 answers

If the roots of quadratic equation are r and s, what is the value of r-s

Let the roots of quadratic equation $x^2 + px + q$ be equal to $r$ and $s$. Using Vieta's formulas, what is the value of $r - s$ in terms of $p$ and $q$?
John Liu
  • 419
  • 3
  • 11
0
votes
1 answer

Quadratic equation, cannot solve it while using a technique that doesn't use bhaskara

My teacher taught how to solve squared equations without bhaskara. It's a completely new technique for me. Actually I solve only 2 problems till now. I am having a hard time solving the last one of the exercise list. Below I will show where I am…
Diego Alves
  • 123
0
votes
2 answers

Intermediate Quadratic Equations

If $n$ is a constant and if there exists a unique value of $m$ for which the quadratic equation $x^2 + mx + (m+n) = 0$ has one real solution, then find $n$. Let the roots of the quadratic be $r,s.$ Vieta gives $-m=r+s, m+n=rs.$ Thus, $n=rs+r+s…
Fleccerd
  • 267
0
votes
1 answer

How to systematically find roots of $x^2-x-132 = 0$ with Po-Shen Loh's method?

Po-Shen Loh in his famous video shows how to systematically find quadratic equation's roots. He find the roots for following quadratic equation. $x^2-8x+12 = 0$ Product: 12, Sum: 8 He divides sum by half i.e. 8/2 = 4 Proceeds to find roots: $4…
Ubi.B
  • 340
0
votes
2 answers

Figuring out coefficients of composition of a first degree polynomial into a quadratic

given $g(x)=x^2 + x-2$ and $ g \circ f = 2 [2x^2-5x +2]$, find $f(x)$ ( $f(x)$ is form $ax+b$) I found the inverse of g as two functions, $$ y =( x + \frac12)^2 - \frac94 $$ $$ \pm (\sqrt{y + \frac94} -\frac12 ) = x$$ or, $ g^{-1} (y) = \pm…
tryst with freedom
  • 11,538
0
votes
2 answers

Invariance of quadratic equation under a horizontal shift?

Suppose, we have a quadratic equation $Q(x)$ and, we shift the input by $\alpha$ i.e: the function is now $Q(x -\alpha)$, such that $Q(x) = Q(x -\alpha)$ Suppose, $$ Q(x) = ax^2 + bx +c$$ and, $$ Q(x-\alpha) = a(x-\alpha)^2 + b(x-\alpha) +c$$ Since,…
tryst with freedom
  • 11,538
0
votes
1 answer

Solving a specific quadratic function for $x$ and $y$

Given are $$S_1 = x(a_1x+2h_1y)+b_1y^2 + c_1=0$$ $$S_2 = x(a_2x+2h_2y)+b_2y^2 + c_2=0$$ $$S_1=S_2$$ I want to solve for a positive $x$ and positive $y$. There is a method as explained at the very end in this source, here's the quote: Each of these…
Phy
  • 153
0
votes
1 answer

The values of $a$ and $b$ for integral $x$

If roots of the equation $x^{2} + ax + b = 0$ are positive integers and $a + b = 198$ then values of $a$ and $b$ are respectively. My solution: Positive real roots means $D > 0$. Therefore $b^{2}-4ac>0$. On putting value of $b$ from $a+b$ gives…
Shaurya Goyal
  • 449
0
votes
0 answers

Solve a quadratic with 2 unknowns

If I have a quadratic $Ax^2 + Bx + C$ where the coefficients contain an additional unknown $u$ for $Ax^2+(bu+c)x+(du^2+eu+g)$ where $A$, $b$, $c$, $d$, $e$, $g$ are known Is there a way to find solutions for $x$ and $u$ that satisfies $0 =…
Blindman67
  • 129
0
votes
3 answers

What is the equation of the quadratic function whose vertex of the graph is on the $x$-axis and passes through the two points $(1,4)$ and $(2,8)$?

What is the equation of the quadratic function whose vertex of the graph is on the $x$-axis and passes through the two points $(1,4)$ and $(2,8)$? Here is my attempt: Use the vertex form of a parabola $f(x) = a(x – h)^2 + k$ where point $(h, k)$ is…
AYA
  • 578
1 2 3
…
64 65