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





{\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:


















{\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.

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  1. DaveE

    An engineer's approach to the quadratic formula

    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...
  2. lilyhachi

    Proving Roots: Formula for Solving Quadratic Equations

    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 +...
  3. C

    Finding a domain for a function

    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##...
  4. A

    Finding the limits of integration and quadratic formula

    Homework Statement Please look at the photo! Homework Equations -x^2+4x=x^2-6x+5 The Attempt at a Solution I got 2x^2-10x+5 but it says it's wrong
  5. C

    Using quadratic formula to find time [projectile motion]

    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...
  6. David23454

    Solve equation using quadratic formula

    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...
  7. stevendaryl

    Insights Solving the Cubic Equation for Dummies - Comments

    stevendaryl submitted a new PF Insights post Solving the Cubic Equation for Dummies Continue reading the Original PF Insights Post.
  8. caters

    Complex zeros to polynomial

    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...
  9. S

    MHB Applying quadratic formula to inverse of a function: Calculation problem

    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...
  10. D

    Finding the quadratic equation that suits the solutions

    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...
  11. M

    Solving GRE Math Question: Quantity A vs B

    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...
  12. S

    As accurately as possible draw a curve in the real space R2

    -3x2 - 8x*y + 3y2 = 1 . Specifie also the axis of the given curve.
  13. binbagsss

    Tricky Quadratic formula / trig identities

    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...
  14. T

    Finding root of complex equation

    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...
  15. B

    A Generalized Quadratic Formula

    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...
  16. J

    About quadratic formula

    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}## ?
  17. X

    Deriving the quadratic formula.

    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...
  18. S

    Understanding the Derivation of the Quadratic Formula

    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...
  19. D

    MHB Are these lines parallel, perpendicular, or neither?

    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...
  20. K

    Quadratic Formula Evaluation

    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.
  21. C

    Kinematics: when to use the quadratic formula?

    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...
  22. MarkFL

    MHB Two methods for deriving the quadratic formula that I was not taught in school

    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...
  23. L

    Quadratic formula in Z_n

    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...
  24. B

    What are some interesting ways of proving the quadratic formula?

    Without the classic method of "completing the square" and factorising?
  25. R

    Quadratic Formula (Equilibrium Points)

    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...
  26. K

    Solve using quadratic formula?

    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...
  27. B

    Please Critique my Presentation of the Quadratic Formula

    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...
  28. K

    Use quadratic formula to solve a function

    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...
  29. agentredlum

    Questions about quadratic formula

    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...
  30. D

    How to find the root of a quadratic formula

    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...
  31. A

    Flaw in the quadratic formula

    But seriously, why am I getting the wrong answer here: 4x^2+10x= 5 4x^2+10x - 5 = 0 x=\frac{-10+\sqrt{180}}{8} Answers Incorrect: 3.461 and -23
  32. A

    Clearing fractions and setting up for the quadratic formula?

    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?
  33. N

    How do you use the Quadratic Formula to solve First Order ODE?

    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...
  34. T

    Kinematics and quadratic formula question

    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...
  35. M

    Prove the quadratic formula

    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...
  36. A

    Solve for x in terms of y (Quadratic formula)

    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...
  37. P

    Rearranging the Projectile Motion Base Quadratic Formula for Initial Velocity

    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...
  38. M

    Quadratic Formula Homework: Meaning of A, B & C in Y = At^2 + Bt + C

    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?
  39. MacLaddy

    Quadratic Formula examples

    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...
  40. S

    Quadratic formula question

    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...
  41. F

    Apply the quadratic formula

    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
  42. M

    Solving Quadratic Equation: Find x for k=3000, m=5, g=9.81, d=1.3

    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?
  43. Y

    Do you need to use the quadratic formula to solve for the unkown?

    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 ?
  44. R

    Using Quadratic formula in other fields

    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?
  45. E

    Complex Quadratic Formula

    [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...
  46. Holocene

    Quadratic Formula: Positive or Negative Root?

    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...
  47. S

    The quadratic formula and equilibrium problems

    can i always the quadratic forumula when i m doing these types of problems. my teacher said i can but sometimes theirs faster ways to do it
  48. S

    Quadratic formula problem

    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...
  49. M

    Velocity using quadratic formula- NEED HELP PLEASE

    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...
  50. C

    Factoring with the quadratic formula

    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...