In algebra, a quadratic equation (from the Latin quadratus for "square") is any equation that can be rearranged in standard form as
a
x
2
+
b
x
+
c
=
0
{\displaystyle ax^{2}+bx+c=0}
where x represents an unknown, and a, b, and c represent known numbers, where a ≠ 0. If a = 0, then the equation is linear, not quadratic, as there is no
a
x
2
{\displaystyle ax^{2}}
term. The numbers a, b, and c are the coefficients of the equation and may be distinguished by calling them, respectively, the quadratic coefficient, the linear coefficient and the constant or free term.The values of x that satisfy the equation are called solutions of the equation, and roots or zeros of the expression on its left-hand side. A quadratic equation has at most two solutions. If there is no real solution, there are two complex solutions. If there is only one solution, one says that it is a double root. A quadratic equation always has two roots, if complex roots are included and a double root is counted for two. A quadratic equation can be factored into an equivalent equation
a
x
2
+
b
x
+
c
=
a
(
x
−
r
)
(
x
−
s
)
=
0
{\displaystyle ax^{2}+bx+c=a(x-r)(x-s)=0}
where r and s are the solutions for x. Completing the square on a quadratic equation in standard form results in the quadratic formula, which expresses the solutions in terms of a, b, and c. Solutions to problems that can be expressed in terms of quadratic equations were known as early as 2000 BC.
Because the quadratic equation involves only one unknown, it is called "univariate". The quadratic equation contains only powers of x that are non-negative integers, and therefore it is a polynomial equation. In particular, it is a second-degree polynomial equation, since the greatest power is two.
In a certain type of problem, a quadratic equation is formed with the square root of energy being the variable to be found ex: (a*sqrt(E)^2+b*sqrt(E)+c=0). Then they claim since energy (E) is real and positive, only solutions to the quadratic equation in sqrt(E ) being real and positive are...
Homework Statement
Find the number of solutions of the equation $$\sqrt {x^2}-\sqrt {(x-1)^2} + \sqrt {(x-2)^2}=\sqrt {5}$$
Answer given: 2
Homework Equations
The Attempt at a Solution
Completely clueless as to where to start.
Homework Statement
P(x) =ax2+bx+c where a, b and c are in arithmetic progression and are positive. α and β are the roots of the equation and are integers. Find the value of α+β+αβ. (Answer is 7)
Homework Equations
x = {−b ± √(b2 − 4ac)} /2a
3. The Attempt at a Solution [/B]
Since a, b and c...
Let A and B be the roots of the quadratic equation
ax^2 + bx + c = 0. Verify each statement below.
1. A + B = -b/a
2. AB = c/a
I need help getting started for parts 1 and 2. I will do the math.
<< Mentor Note -- thread moved from the technical forums, so no Homework Help Template is shown >>
Saw this problem the other day and I have a question about the solution(s):
A river is flowing downstream at a speed of 3 mph. A boat travels up the river 24 miles, turns around and travels down...
Homework Statement
How to solve for kd or equation with 2 unknowns given:
b= 10
n= 8
As = 2.37
Homework Equations
[/B]
for the quadratic equation b * (kd)^2 /2 - n * As (d - kd) = 0
The Attempt at a Solution
should one use algebra, calculus, what? how do you solve for kd? this is not...
HI! I'm not sure if this can go in precalculus or not because I'm from Australia, and our Maths subjects don't get that specific until university level.
1. Homework Statement
For my assignment on quadratic functions, I have to find the equation (the the form of ax^2+bx+c) for a table of...
Is it possible to factor a quadratic equation along the lines of asin^2x -bsin2x+c ? If so, how? The sin2x seems to be a problem since when expanded it becomes 2sinxcosx, but I'm wondering if it is possible, and how it would be done?
Thanks in advance.
Homework Statement
X^2-yx+y^2-7=0
Homework Equations
-b +- sqrt(b^2-4ac)/2a
The Attempt at a Solution
Trying to complete the square with two variables
(x^2-yx)+(y^2)=7
(X^2-yx+y/2)+y^2=7
Where else from here. I'm just having problems because of the y variable
Homework Statement
I'm looking for an explanation to something. I've attached a picture of the solution wolfram alpha is giving me.
I understand the first two zeros, +- (-1)^(1/4)*sqrt(2).
But i don't understand the other two zeros with the 3/4 power. Where does that power come from...
Homework Statement
Part (b)
Homework Equations
Vieta's formula
The Attempt at a Solution
Part (a):
I'm kinda stuck on part (b) - I tried multiplying the product of roots and got b^2 d^2, but I have no idea on dealing with the sum of roots.
1. Homework Statement Homework EquationsThe Attempt at a Solution
I just want some clarification about the rules on solving quadratic equation. My question is coming from my solution for problem 49.
Solving problem 49 [/B]
Factor: (x−p−q)(√(x−q)+√(x−p))=0
Therefore: x−p−q=0⇒x=p+q this...
Hello! I just want to clarify something about the rules on solving quadratic eqns.
I have already solved both problems but in the process of my solution some questions arise.
For example in the part of my soltution in problem 48 I have this...
A person, selling a horse for $72, finds that his loss per cent is one-eight of the number of dollars that he paid for the horse; what was the cost price?
Can anybody explain the part " loss per cent" and how do I express that algebraically. Thanks!
Hello! I just want to solve this exercise using shortest way possible.
$\frac{k-n}{2m+x}+\frac{m-n}{2k+x}=\frac{k+m-2n}{k+m+x}$
Because when I tried it is so lengthy. Maybe you can teach me how to go about it faster. Thanks
a man spent 78 dollars for cigarettes. has the price per box been .50 cents less, he could have had one more box. How many boxes did he buy?
Heres what I tried
let $y=$ original price per box
$y-50=$ new price per box
Now,
$\frac{780}{y}=$ original number of boxes bought...
Homework Statement
Showing all necessary working solve the equation ##iz^2+2z-3i=0## giving your answer in the form ##x+iy## where x and y are real and exact,Homework EquationsThe Attempt at a Solution
##iz^2+2z-3i=0, z^2+(2/i)z-3=0##,using quadratic formula →##(-2/i± √8)/2 , z= √2+1/i## and...
Hello I've been stuck on this test review question for a few days, and I can't figure it out. Can someone help out?
"3x²+12x+c=0, Find solutions for c, where there is 1 real solution, 2 real solutions, and 2 nonreal(complex) solutions"
Let $a$ be the smallest root of the equation $x^2-9x+10=0$. Find $a^4-549a$. Extra credit if the solution does not find the actual roots of the equation.
$\alpha$ and $\beta$ are the roots of the equation $2{x}^{2}-5x+c=0$. If $4\alpha-2\beta=7$, find the value of $c$.
I did the following:
$\alpha+\beta=-\frac{-5}{2}=\frac{5}{2}$
$\alpha\beta=\frac{c}{2}$
$\frac{c}{2}=\frac{7+2\beta}{4}\cdot\frac{-7+4\alpha}{2}$...
Homework Statement
if x1 and x2 are solutions to 1) x² + x + 1 = 0, then 2) y1 = ax1 + x2 and 3) y2 = x1 + ax2 are solutions to which quadratic equation?
Homework Equations
ax² + bx + c = 0
x1∕2 = (−b ± √(b² - 4ac))/2a
The Attempt at a Solution
Well, firstly i solved for x1 and x2 getting...
Is possible classify the quadric equation Axx + Bxy + Cyx + Dyy + Ex + Fy + G = 0 how straight, hyperbola, circle, ellipse, parabola, etc, in the same way that is did in the phase plan:
https://upload.wikimedia.org/wikipedia/commons/3/35/Phase_plane_nodes.svg...
Not homework but given the question it probably fits here best
Given the following equation
$$x^2+138x+317=y^2$$
How do you find the integer solutions?
For example wolframalpha has the solutions. but I cannot see how they are derived
http://www.wolframalpha.com/input/?i=x^2+138x+317=y^2
By...
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >
Hi,
I'm having difficulty solving the following quadratic equation
x^2-(660/x-5)-91=0. The fact that the middle term has an x-5 rather than just an x that is throwing me
Can anyone please help...
Homework Statement
The Attempt at a Solution
Looks like the graph would be a parabola?
And since it's a>0 it would be upward and therefore a minimum. Not sure what that would be. Unsure how to solve the rest[/B]
Homework Statement
3x^2 + px + 3 = 0, p>0, one root is square of the other, then p=?? [/B]Homework Equations
sum of roots = -(coefficient of x)/(coefficient of x^2)[/B]
product of roots= constant term/coeffecient of x^2
The Attempt at a Solution
ROot 1 = a
root 2 = b
b=a^2
a.b= 3/3
a^3=1...
Homework Statement
A functions is defined as f(x) = ax2 + bx + c, where a, b, c are real numbers. If f(3) = f(– 2) = 0, what is the value of f(0)?Homework EquationsThe Attempt at a Solution
As function is 0 at 3, -2, therefore
9a + 3b + c=0 also,
4a -2 b + c=0
c=-6a or c= 6b
f(0)=c= 6a...
Homework Statement
First part of the question (SOLVED)
A football player kicks a football so that the angle of incidence is 50 DEG and the initial magnitude of velocity of the ball is 15 m/s.
Find the:
a) Balls maximum height = 6.7 m
b) Time of flight = 2.3 s
c) time when the ball reaches...
The equation is: (appears while solving a trigonometric integral using residue theorem)
2Z2+iZ2-6Z+2-i
=(2+i)Z2-6Z+(2-i)
The roots are:
Z1=(2-i) and Z2=(2-i)/5
I can't write the equation in factored form.
If I simply write it like this:
{z-(2-i)}{5z-(2-i)}
It doesn't give the same...
Hi, first off I want to say that I'm new here, so sorry if I do anything wrong.
Okay, now to the problem at hand. I know that this is probably really easy and I'm just having one of my moments again, but I can't for the life of me figure out how to do this question:
When the radius of a...
Definition/Summary
A second order polynomial equation in one variable, its general form is ax^2 + bx + c = 0, where x is the variable and a, b, and c are constants, and a \ne 0.
Equations
ax^2 + bx + c = 0
Extended explanation
Since a quadratic equation is a second degree...
Homework Statement
The number of integral values of 'a' for which the quadratic equation (x + a) (x + 1991) + 1 = 0
has integral roots are:
Homework Equations
D = b² - 4ac
The Attempt at a Solution
What I did was simplify the given equation and I got:
x² + (1991 + a)x + (1991a + 1) = 0...
I'm having problems getting going on the following question, any help appreciated:
As part of his training an athlete usually runs 80 km at a steady speed of \(v\) km h. One day he decided to reduce his speed by 2.5 km h and his run takes him an extra 2h 40 mins.
Derive the equation...
Question 1
(a) Solve the quadratic equation
x^2 + 4x -5 = 0
(b) Factorise its left hand side.
(c) Find interval(s) of x where the left hand side is positive
Q1
(a)
(x+-1) (x+5)
x= 1 x=-5
is that solving the equation?
(b)
didnt i already factorise the left hand side...
Can a quadratic equation with rational coefficients have one rational root and one irrational root? explain.
and
Can a quadratic equation with real coefficients have one real
root and one imaginary root? Explain.
please enlighten me.
Please help me continue this problem
A market vendor bought a crate of mangoes for 55 peso. when the crate was opened he found that 4 pieces were not fit to be sold. If he sells the rest at 80 cents more than the buying price, he gets a profit of 8 peso for these remaining mangoes. How many...
please check my work.
Pipe A can fill a given tank in 4 hr. If pipe B works alone, it takes 3 hr longer to fill the tank than if pipes A and B act together. How long will it take pipe B working alone?
let $x=$ required time for B working together with A
$x+3=$ required...
please help me with this
$\frac{x^2+2}{x}+\frac{8x}{x^2+2}=6$
this is where I can get to when I simplify the the equation above,
$x^4-6x^3+12x^2+12x+4=0$
can you show me a way of solving this problem without considering the discriminant.
Find the roots of equation subject to the given condition.
$(2m + 1)x^2-4mx = 1-3m$ has equal roots.
I solved it using discriminant but I want to know other way of solving it. Thanks!
please check my answers if they are correct. these problems are even numbered probs in my books that's why I need somebody to check it.
1. solve for x in terms of other symbols
$x^2-2xy-4x-3y^2=0$
using the quadratic formula I get
$x=y+2\pm4\sqrt{y^2+y+1}$
2. what is the width of a strip...
please help me get started with these problems.
1.) It took a faster runner 10 sec longer to run a distance of 1500 ft than it took a slower runner to run a distance of 1000 ft. If the rate of the faster runner was 5ft/sec more than the slower runner, what was the rate of each?
2.) It is...
If $a,b,c$ are the length of the sides of an scalene triangle, If the equation
$x^2+2(a+b+c)x+3\lambda\left(ab+bc+ca\right) = 0$ has real and distinct roots,
Then the value of $\lambda$ is given by
Options::
(a) $\displaystyle \lambda < \frac{4}{3}\;\;\;\;\; $ (b)$\displaystyle...
If the quadratic equation $x^2+(2 – \tan \theta)x – (1 + \tan \theta) = 0$ has two integral roots, then sum of all possible values of $\theta$ in the interval $(0, 2\pi)$ is $k\pi$. Find $k$.