What is Quadratic equations: Definition and 98 Discussions
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.
ok this was posted on LinkedIn and sure it has already be answered
but usually these types of problems are resolved by way too many steps
so just wanted to proceed with this without looking at previous attempts
my first reaction was to get a CD but would introduce a bigger problem
however...
Hello:
I'm not sure if there's an accepted canonical form for a quadratic equation in two (or more) variables:
$$ax^2+by^2+cxy+dx+ey+f=0$$
Is it the following form? (using the orthogonal matrix Q that diagonalizes the quadratic part):
$$ w^TDw+[d \ \ e]w+f=0$$
$$w^TDw+Lw+f=0$$
where
$$...
Given equation and conditions: ##\boldsymbol{x^2+2(k-3)x+9=0}##, with roots ##\boldsymbol{(x_1,x_2)}##. These roots satisfy the condition ##\boldsymbol{-6<x_1,x_2<1}##.
Question : ##\text{What are the allowable values for}\; \boldsymbol{k}?##
(0) Let me take care of the determinant first...
On simplifying the given equation we get, x^2-x-1=0 and using the quadratic formula we get x=(1+√5)/2 and x=(1-√5)/2
Now, as the formula suggests, there are two possible values for x which satisfies the given equation.
But now, if we follow a process in any general calculator by entering...
Given are two equations:
$$S_1 = ax^2+2hxy+by^2 + c=0$$
$$S_2 = a'x^2+2h'xy+b'y^2 + c'=0$$
This source states that there are several methods to solve for ##x## and ##y##. One of them is the following quote:"Treat equation ##S_1## as a quadratic equation in ##x## and solve it for ##x## in terms...
I have 2 quadratic functions and I am interested in their root in the specific range. I use quadratic equation to get their roots and what I find that if their any real solution exist for both or any of the function that lie in it designated specific range, then the roots are maximum or minimum...
−2x^2+3x+20why is this equation so special it is in standard form, and when i solve for zeros
i factor down to
-1( 2x^2-3x-20)
factor
( 2x^2-3x-20)
factor by grouping 2(-20)=-40 what numbers multiple to equal -40 and add to equal -3
-8(5) = -40
-8+5= -3
and then just to find zeros the...
Homework Statement
If ##x## is a rational function of ##y##, such that (ax2 + bx + c)y + (a'x2 bx' + c') = 0
prove that (ac' - a'c)2 = (ab' - a'b)(bc' -b'c)
Homework Equations
The quadratic formula
The Attempt at a Solution
This equation can be rewritten as:
(ay + a')x2 (by + b')x (cy +c') = 0...
Homework Statement
If ax2 - bx + 5 = 0 does not have two distinct real roots, find the minimum value of 5a + b.
2. Homework Equations
The Quadratic Formula
The Attempt at a Solution
Here, D = b2 - 4a(5) = b2 - 20a
D ≤ 0 ⇒ b2 ≤ 20a ⇒ b ≤ ±2√(5a)
Also, an obvious observation is that b ≤ 0...
I would like to solve this system, which is a sets of non linear quadratic equations, the system needed to be solved can be expressed in general as follow:
ϒϒ'C – ϒα = B
Where ϒ=(ϒ1,ϒ2,...ϒn)’ is a column vector and ϒ’ its transpose
C=(c1,c2,…,cn)’ and B=(b1,b2,…bn)’ are a columns vector
And...
Hello everyone! Apologies if this is a very repetitive question but I have gone through previous forum posts and am still struggling to understand how to identify which equations are appropriate. In the problem below, I have used the kinematic equation of "v = v(i) + at" but my answer is...
Homework Statement
1)The value of k, so that the equations 2x2+kx-5=0
and x2-3x-4=0 have one root in common
2)The value of m for which one of the roots of x2 is double of one of roots of x2-x+m=0
3)If x2-ax-21=0 and x2-3ax+35 have a root in commom
Homework EquationsThe Attempt at a Solution
I...
Homework Statement
Homework Equations
for equation which has 2 different solutions, D >0
The Attempt at a Solution
(1)[/B] D > 0
b^2 - 4ac > 0
3 - 4root2.k > 0
k < 3 / ( 4root 2 )
k < ( 3 root 2 ) /8
has solution of sin tetha and cos tetha
sin 0 = 0, cos 0 = 1.
when x = 0, and x = 1 -->...
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.
Consider the quadratic equation x^2+px+2p=0
a. Find the discriminant.
b. Find the values of p for which there are 2 solutions.
c. Find the values of p for which there are no solutions.
d. Find the value of p for which there is 1 solution.
Please show working out! Thanks.
Ian in London (South).Interests Number Theory. Prime Numbers...
Integer iterations of this quadratic expression only yield decimal numbers whose factors end with the digit _1, or _9. Why?
Homework Statement
Let ##a,b,c## be positive integers and consider all the quadratic equations of the form ##ax^2-bx+c=0## which have two distinct real roots in ##(0,1)##. Find the least positive integers ##a## and ##b## for which such a quadratic equation exist.
Homework EquationsThe Attempt...
I derived a relation between the product of two linear polynomials and the square of their average. It can be used to solve any quadratic equation. Will this help me getting into a top university?
Homework Statement
[/B]
Th value of 'a' for which the equation x3+ax+1=0 and x4+ax+1=0 have a common root is?Homework EquationsThe Attempt at a Solution
i initially thought of subtracting both the equations and then finding x and substituting back in the equation but it did not work.
so I've been tasked with this insane question
obtain the solution of the following quadratic equation
how do I solve this, the 'E' (greek letter),is throwing me out,my complete hazardous guess is
2E(S+Wn)2 <--- 2 is a squared sorry my keyboard skills are lacking also 'tut'
any help...
Homework Statement
Homework Equations
Not Sure.
The Attempt at a Solution
For the first question I know you have to multiply the conjugate of the denominator so it would be (2 - √5)/(1−2√5) x (1−2√5)/(1−2√5) but I'm not sure how to actually do that.
For the second question. I have that...
A train travels a distance of 480km at a uniform speed. If the speed had been 8km/h less, then it would have taken 3 hours more to cover the same distance. Find the speed of the train.
Represent this situation in the form of quadratic equation.
So I have been trying this but couldn't find...
I understand how to solve these equations when the square is on this side of the equal sign: x2 + 8x + 7 = 27
But when the square is on the other side, I am thrown. Like this one...
x2 = 14x - 33
The solutions manual shows the next step as the following, but what do you do to get to this...
Homework Statement
If roots of the equation x^2 - (2n+18)x - n - 11 = 0 (n is an integer) are rational for n=a and n=b then |a-b| is
Ans: 3
Homework EquationsThe Attempt at a Solution
On substituting a (or b) into the quadratic, the roots are rational.
If the roots are rational, then the...
Homework Statement
If a, b, c, d are the positive roots of x^4 - 12x^3 - px^2 + qx + 81 = 0, then the area of the quadrilateral formed by x=a, x= -b , y = c and y = -d is:
Ans: 36
Homework Equations
Vieta, I guess.
The Attempt at a Solution
I know its going to form a rectangle with sides...
I get why it's a parabola because of the x^2 (for every value of x, y is the square of that number), but why does it shift to the left (and down as well) when I add x?
Homework Statement
Let ##a,b,c## and ##m \in R^{+}##. Find the range of ##m## (independent of ##a,b## and ##c##) for which at least one of the following equations, ##ax^2+bx+cm=0, bx^2+cx+am=0## and ##cx^2+ax+bm=0## have real roots.Homework Equations
The Attempt at a Solution
I don't really...
Homework Statement
The following enclosure is built using 280 m of fencing. If the enclosure has a total area of 2800 m2, what are the dimensions to the nearest tenth?
Homework Equations
x = \frac{-b ± \sqrt{b^{2}-4ac}}{2a}
The Attempt at a Solution
Please note that the 3/2w and 5/2w...
Homework Statement
Let a,b,c be real numbers with a>0 such that the quadratic equation ##ax^2+bcx+b^3+c^3-4abc=0## has non real roots. Let ##P(x)=ax^2+bx+c## and ##Q(x)=ax^2+cx+b##. Which of the following is true?
a) ##P(x)>0 \forall x \in R## and ##Q(x)<0 \forall x \in R##
b) ##P(x)<0...
Hi, I've been attempting to teach myself mathematics in preparation for further study and I'm currently going through quadratic equations. In case it helps, I'm studying to go back to university, I'll be doing a physics degree with a foundation year. I'm just trying to learn as much math as...
I have been reading up on Quadratic Equations in my course book, which gives some examples of simple equations and how they are factorised, but can't get my head round this type which is has no examples.
x2 - 3x = 0
OK I know it is a quadratic because the term x2 is included. What I require...
Hi,
This is perhaps the most rookie question to Mathematics that one could ask, but I have searched for information on the question and found only one source which contained an answer. The answer was that it is because otherwise we wouldn't know what the value of either of the factors is. I...
Could anyone please verify with me that I have the right idea in answering the question below.
Homework Statement
If you know the initial velocity v0 and the initial and final heights y0 and y, you can use x=x0+v0t+(1/2)at^2 to solve for the time t when the object will be at height y. But...
Well suppose for an example of an inequality,
|x-1|-|x|+|2x+3| > 2x+4
Well in one of its solutions we were told to apply the method of intervals, rather than taking say what; like 8 combination of signs.
For everyone of its intervals(say -3/2 \leqx <0) we are said that 2x+3 \geq 0, x<0...
theory of quadratic equations...
theres a quadratic equation lx^2 + nx + n where its roots are in the ratio p:q .we need to prove that
√(p/q) + √(q/p) + √(n/l) = 0
what i did was..i introduced a proportionality constant k... so pk + qk = -(n/l)
while pq(k^2) = n/l...
I'm sorry, I just realized I put this in the wrong subsection. While I figure out how to fix that, please have a look anyway.
__
Homework Statement
Given x \inℝ
And s =\frac{4(x^{2}) + 3}{2x-1}
Prove that s^{2} -4s - 12 ≥ 0
Homework Equations
The discriminant Δ, (in order for which to be...
Homework Statement
IMPLICIT NONE
REAL :: A, B, C, DISCR, X1, X2, x1i, x2i
CHARACTER(3)::ANS,ANS1
! Reads the coefficients for the quadratic equation
33 WRITE(*,*)")"Please enter a *REAL*NUMBER* coefficient ( A )."
READ(*,*)A
WRITE(*,*)"Please...
1. The problem:
Find the value of \lambda for which the sum of squares of the roots of the equation: 2x^2 + (2\lambda +4)x^2-(1+ \lambda) = 0 has minimum value.
2. Homework Equations
\alpha + \beta = -b/a
\alpha\beta = c/a
where a,b,c are coefficients of x^2 , x and constant...
A quadratic equation has the form y = ax^2 + bx + c. However, if c = 0, then y = ax^2 + bx. Is it still called a quadratic equation? And if b = 0 so that y = ax^2, is it still given the title of quadratic equation?
I would guess yes since it still has a power of 2 and is a parabola. Is...
Hello PF! I'm having trouble approaching this problem. Any assistance would be greatly appreciated.
Homework Statement
A rectangle with area of 35 cm2 is formed by cutting off strips of equal width from a rectangular piece of paper. The rectangular piece of paper is of 7cm width and 9cm...
Hi guys,
I'm really confused in finding the domain of quadratic equations.
For example: when finding a suitable domain so that an inverse exists, why is the domain of x2-4 x>0
whilst,
the domain of 2x2+3 is x≥0
Can the domain of x2-4 be x≥0?
Furthermore, what is the largest domain and how do...