What is Quadratic formula: Definition and 59 Discussions
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others.Given a general quadratic equation of the form
a
x
2
+
b
x
+
c
=
0
{\displaystyle ax^{2}+bx+c=0}
with x representing an unknown, a, b and c representing constants with a ≠ 0, the quadratic formula is:
where the plus-minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become:
x
1
=
−
b
+
b
2
−
4
a
c
2
a
and
x
2
=
−
b
−
b
2
−
4
a
c
2
a
{\displaystyle x_{1}={\frac {-b+{\sqrt {b^{2}-4ac}}}{2a}}\quad {\text{and}}\quad x_{2}={\frac {-b-{\sqrt {b^{2}-4ac}}}{2a}}}
Each of these two solutions is also called a root (or zero) of the quadratic equation. Geometrically, these roots represent the x-values at which any parabola, explicitly given as y = ax2 + bx + c, crosses the x-axis.As well as being a formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of symmetry of the parabola, and the number of real zeros the quadratic equation contains.
I stumbled across this article from decades past written by the best EE instructor I ever had. I thought it might be of some passing interest to others in highlighting the difference between memorizing a mathematical result vs. truly understanding it. The essence of engineering, in effect. We...
Summary:: Hi guys, i can't seem to get the correct answer. I'm wondering where did I do wrong. Can someone help me to solve this? I think I need the correct formula to prove the answer :(
Given a root to 𝑥² + 𝑝𝑥 + 𝑞 = 0 is twice the multiple of another. Show that 2𝑝² = 9𝑞. The roots for 𝑥² +...
I am having some trouble find the domain with this function: ##f(x)=\frac{1}{\sqrt{x^2-4x\cos(\theta)+4}}## and ##\theta\in[0,\pi]##.I know that the denominator needs to be greater than 0. So ##\sqrt{x^2-4x\cos(\theta)+4}>0##. I squared both side of the inequality, ##x^2-4x\cos(\theta)+4>0##...
1. Homework Statement
I'm doing lab, and to start we have to find ΔX of a projectile launched an angle. The first step for us is to find time, and to make it easier our teacher recommended us to first set the angle to zero, and find time like that.
I don't have enough X information to find...
Homework Statement
I'm having trouble with solving a certain form of an equation with the quadratic formula. I think I'm making a dumb mistake somewhere with my algebra, but I can't seem to find it.
Solve with quadratic formula: 1/x^2=4/(0.02-x)^2
Homework Equations
1/x^2=4/(0.02-x)^2
The...
Homework Statement
Form a polynomial whose zeros and degree are given below. You don't need to expand it completely but you shouldn't have radical or complex terms.
Degree 4: No real zeros, complex zeros of 1+i and 2-3i
Homework Equations
(-b±√b^2-4ac)/2a
The Attempt at a Solution
I want...
Hi - I'm trying to figure out the specific steps in a math textbook. I'm trying to figure out how the textbook did its algebra here with a quadratic formula to find an inverse of a function:
Sadly there's a step I'm supposed to know by now in the simplification algebra in there 3 steps...
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...
I was going through some GRE Quantitative Practice Tests, and found this question. Maybe I'm not understanding the question right (not a native speaker, and have never solved math problems in English), but choice B doesn't seem correct to me.
Solutions for x (Quantity A) are 6 and -1, and for...
Im to solve ##(k+l)^{2}e^{-ila}-(k-l)^{2}e^{ila}=0##, for ## k##,
The solution is ##k=l(e^{ial}-1)/(e^{ial}+1)=il tan(al/2)##
FIRST QUESTION
So it's a quadratic in k, should be simple enough, my working so far using the quad. formula is ##k= (4l^{2}(e^{-ila}+e^{ila})\pm...
Homework Statement
Good day,
I've been have having difficulties finding the roots of this:
Find the roots of 3ix^2 + 6x - i = 0
where i = complex number
i = sqrt(-1)
Homework Equations
quadratic formula (apologies for the large image)
The Attempt at a Solution...
Just over two years ago, I was introduced to the process of completing the square as a way to solve the roots in a quadratic equation. More recently, I've thought about how I could go about extending this to completing the cube. The following short story/proof is the result of during just that...
The solutions ##(x_1, x_2)## for the quadratic equation ##(0=ax^2+bx+c)##:
##x_1 = \frac{-b + \sqrt{b^2-4ac}}{2a}##
##x_2 = \frac{-b - \sqrt{b^2-4ac}}{2a}##
Are true if ##x## and ##a##, ##b##, and ##c## ##\in## ##\mathbb{C}## ?
1.Homework Statement
Derive the quadradic equation, ax^2 + bx + c = 0, using the fallowing method.
1. Divide by a, if a =/ 0.
2. Move the constant to the other side of the equation.
3. Square half the coeffecient of bx and add it to both sides, fixate the negative sign on -c/a.
4. factor the...
This could be seen as a rather "basic" math question, but it is a topic of curiosity for me. I'm currently a senior in high school, taking a pre-ap pre-cal/trig/AP-Calculus double blocked class. I'm absolutely fascinated by mathematics, and something of keen interest to me is the derivation of...
Basically I don't know anyone in real life that can help me with this, so I need help checking to see if my answers are correct :)
PART A
3) Solve 2x^2 + 4x = 15 by using the quadratic formula.
x = -1 +/- 2sqr34
4) Determine whether the given pairs of lines are parallel, perpendicular, or...
Could someone please help with this evaluation?
Homework Statement
-1+/-√11.24/-0.0256
The solutions given for this are -91.9 and 170 and I am truly flabbergasted as to how this instructor achieved these values.
Hi PF, I'm reviewing my notes from class, starting from the very beginning. I'm working on some problems using kinematic equations. Here's one example: a ball falls from 30m (using down as the positive direction) at a velocity of 8 m/s. how long does it take the ball to hit the ground?
Why do I...
As a student, I was taught 3 ways to solve quadratic equations:
i) Factoring
ii) Completing the square
iii) Applying the quadratic formula, derived by completing the square on the general quadratic in standard form:
(1) $\displaystyle ax^2+bx+c=0$
To complete the square, I was taught to...
Working through A Book of Abstract Algebra, I encountered several exercises on roots of polynomials in Z_{n} I was just wondering whether there exists something like a quadratic equation for polynomials of degree 2. If the solutions of the usual quadratic formula happen to be integers, can one...
Homework Statement
The following is part of a solution to a problem about finding equiblirum points of a differential equation and sketching its bifurcation diagram
http://img32.imageshack.us/img32/9194/63226925.jpg [Broken]
How did they get y= -1 \pm \sqrt{1- \mu}?
Homework Equations
The...
Homework Statement
Hey guys, recently, I struggled to solve this equation.
I need to find θ1 and θ2 by using this system of equations:
θ1+θ2=γ
θ1*θ2=β
After letting θ2=β/θ1, I plugged this into the first equation. I got:
θ12-γθ1+β=0
At this point I get a very unsexy θ1 result by using...
Hi, Everyone: I have a job interview tomorrow where I must give a 10-min presentation on the quadratic formula for an intro class , where we are assuming students know both how to factor and how to complete the square. Please comment:
Homework Statement
O.K. We are given an equation...
Homework Statement
L-1(1-(Lambda1+Lambda2)L+Lambda1Lambda2L2)Ft
Use quadratic formula to solve for Lambda1 and Lambda2
The Attempt at a Solution
I took the equation inside the brackets and solved for L. It turned out that
L1=Lambda1 and L2=Lambda2
But when I plugged those...
I have 2 questions about -b and - 4ac in the formula...
x = (-b +-sqrt(b^2 - 4ac))/(2a)
If you are given 2nd degree equations at random (that can be solved using above formula)
Question1: What percent of the equations given would you expect to compute a double negative for -b?
Question2...
Homework Statement
(A) For what value of K does the equation 2x^2-10x +8 = K have a repeated root? Use the quadratic formula to find the root.
(B) Choose a value for K so the equation has no real roots, Use the quadratic formula to prove that this equation has no real roots for your...
How do I clear fractions in problems like this and set it up for the quadratic formula:
\frac{1}{x-1}+\frac{2}{3}=\frac{2}{x-3}
I would like to set this up to be solved with the quadratic formula. Also, is it true that the formula can solve all quadratic equations?
Homework Statement
How do you use the Quadratic Formula to solve First Order ODE?
For example, I am given this integration (see attachment at the bottom). Homework Equations
The Attempt at a Solution
I integrated both sides but I do not know where to go from there (see attachment at the...
Homework Statement
You throw a penny straight up at a speed of 10m/s from a height of 1 m.
1. what's it position 0.5 seconds after it is released? - i found that value to be 4.775 m
2. how high up does it go? = 6.10 m
3. what is its speed when it hits the ground? =?
4. how long is the...
Let F be a field. Suppose f(x)=ax^2+bx+c is an element of F[x] with (2a) a unit in F, and set delta=b^2-4ac. Prove the following version of the quadratic formula. Hint: for (a) and (b) use the identity 4a x f(x)=(2ax+b)^2-delta.
(a) If there is a k in F such that k^2=delta, then (-b plus or...
Homework Statement
Use the quadratic formula to solve the equation for (a) x in terms of y and (b) y in terms of x.
Homework Equations
4x^2 - 4xy + 1 - y^2 = 0
The Attempt at a Solution
I am not really sure where to start at all. If I could just figure out the values for a, b, and...
Homework Statement
What I'm supposed to do is to rearrange the this formula -16t^2+Vt+h and solve it for V.
V= Initial Velocity t= time (throwing the ball in a parabolic arc) h= height
I know what the height is and it is 6ft. I also have the time which is 2.03 seconds.
How do I rearrange...
Homework Statement
What are the meanings of each of the parameters in the Y = At^2 + Bt + C equation?
example:
A = -4.8
B = 21.9
C = -23.8
I know that B is suppose to represent the initial velocity, but not sure about the other 2. Is A supposed to represent acceleration?
Homework Statement
The Quadratic Equation. I am looking for an example of the quadratic equation being used in any form of engineering, or physics.
I am supposed to give an example for my algebra class, but I really don't want to do the "two trains left a station," or "sally jumped off a...
Homework Statement
.002x - .000001x^2 = .50
Homework Equations
-b+-sq.rt.((b^2)-(4ac))/2a
[b]3. The Attempt at a Solution
Plugging a=-.000001, b=.002, and c=-.5 does not get the the correct answer. x is supposed to be 292.89. I can't remember any other way of going about...
Applying it to
-x2 - bx +c = 0
I would have thought it to be
(b +- (b2+4c)0.5) /-2
= (-b +- (b2+4c)0.5) /2
But
Anytime I try and graph a function like it the roots are
(b +- (b2+4c)0.5) /2
I can't see why this is?
Thanks
Using the quadratic formula, solve the distance:
x= distance needed to solve (also our y value in the formula)
A= .5*k
B= -m*g*sin(30)
C= -m* g* d*sin(30)
Where:
k= 3000
m= 5
g=9.81
d=1.3
I get 48.13, but it says that I am wrong. what does everyone else get?
Homework Statement
Lets say you have Ax + Bx^2 + C = 0
Usually I would use the Quadratic formula in this situation.
But what if you do this
x( A + Bx ) + C = 0
x( A + Bx ) = -C
Couldn't you then say
x = -C
and
A + Bx = -C
x = (-C - A) / B
?
Since the the discovery of the complex number field resulted in part from efforts to resolve the quadratic formula when b^2 - 4ac was negative, am I right to use the quadratic formula to solve zM^2 + wM - t = 0 in the complex number field even when z, w or t can be fractional powers?
[SOLVED] Complex Quadratic Formula
Homework Statement
Let a, b and c be complex numbers with a not equal to 0. Show that the solution of az^2 + bz + c = 0 are z_1, z_2 = (-b \pm \sqrt(b^2 - 4ac))/(2a)
The attempt at a solution
I'm assuming z is also complex. Multiplying the equation by...
Regarding \displaystyle{\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}}
How should "\displaystyle{\pm}" be treated?
I know a square root can be both possitive and negative, but does the quardratic forumla dictate that a possitive root should be added to -b, or does it dictate that a postive root...
Homework Statement
A physics student with too much free time drops a watermelon from the roof of a building height H. He hears the sound of the watermelon going "splat" after a time interval of Delta T.
You may ignore air resistance. How high is the building? The speed of sound is Vs...
Velocity using quadratic formula- NEED HELP PLEASE!
Here is the problem:
An osprey is flying over a reservoir with a trout in her talons. Her speed is 4.21 mph and the reservoir is 12.76 feet below. She is flying at an angle of 15.32 degrees below the horizontal-
A- Calculate the...
I have a huge headache and I can't understand why this works... I am reviewing the quadratic formula so that I can solve trig questions, but I am stuck on this one part...
The point is to factor x^2+2x-1
I ( and the textbook ) use the quadratic formula to get the roots of it and come up...