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





{\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




{\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







{\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.

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  1. karush

    MHB Solving Quadratic Equations without CD: Better Direction?

    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...
  2. A

    I Canonical Form for quadratic equations *with* linear terms

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  3. brotherbobby

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    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...
  4. kshitij

    Quadratic equation and its roots

    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...
  5. J

    I Can this particular method solve these quadratic equations?

    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...
  6. chwala

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  7. G

    MHB Quadratic equations intersaction point is minimum instead of roots

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  8. M

    MHB Quadratic equations ( solving for zeros/roots) trickey

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  9. baldbrain

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  10. baldbrain

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  11. T

    Solving a set of nonlinear quadratic equations

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  12. S

    When to use quadratic equations?

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  13. M

    Questions of Quadratic Equations and thier Roots

    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...
  14. H

    How to find a constant in this quadratic equation?

    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 -->...
  15. Wrichik Basu

    A problem in Quadratic Equations

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  16. J

    MHB Quadratics: Quadratic Equations

    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.
  17. Janosh89

    B Why is the quadratic expression 20*x^2-1 only divisible by 11,19,29....

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  18. D

    Got stuck due to the inequality not being satisfied

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  19. Mrq

    Admissions Solving Quadratic Equations with a Linear Polynomial Relation

    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?
  20. V

    What is the common root for two polynomial equations with a shared coefficient?

    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.
  21. F

    MHB Solve Quadratic Equation: 2E(S+Wn)2

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  22. emrys

    Solving Quadratic Equations - (x2+3x+3)1/3 + (2x2+3x+2)1/3 = 6x2+12x+8

    Homework Statement (x2+3x+3)1/3 + (2x2+3x+2)1/3 = 6x2+12x+8 2.Relevant equationsThe Attempt at a Solution (x2+3x+3)1/3 -1 +(2x2+3x+2)1/3 -1 = 6x2+12x+6 (x2+3x+2)/((((x2+3x+3)1/3)2 + (x2+3x+3)1/3 +1) + (2x2 +3x+1)/((((2x2+3x+2)1/3)2)+(2x2+3x+2)1/3 +1) -6(x+1)2=0 then x=-1 or...
  23. M

    Rationalize denominator & factorising quadratic equations.

    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...
  24. manogyana25

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  25. M

    MHB Help with Quadratic Equations by completing the square

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  26. anemone

    MHB Finding Constants for Quadratic Equations

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  27. E

    Solve Quadratic Equations: Find |a-b| for n=a,b

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  28. E

    Quadratic Equations: Area of Quadrilateral Calculation

    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...
  29. J

    Question about how quadratic equations make their graphs

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  30. Albert1

    MHB Proving $1 \leq a \leq 9$ for Quadratic Equations

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  31. Albert1

    MHB Solving Quadratic Equations: Find k

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  32. MarkFL

    MHB Quadratic Equations: Cedric Cajigas' Qs Answered

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  33. Saitama

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  34. S

    Quadratic equations: Perimeter and Area fencing dimensions

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  35. Saitama

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  36. TalkOrigin

    Simple Question Regarding Quadratic Equations (pre-calc)

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  37. C

    MHB Solving Quadratic Equations: Tips & Tricks for Beginners

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  38. K

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  39. S

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  40. K

    Quadratic equations and inequalities

    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...
  41. V

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  42. E

    Maths - a proof question on the nature of roots of quadratic equations

    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...
  43. L

    Comp Sci Solving Quadratic Equations: Invalid Inputs

    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...
  44. T

    Sum of Squares of Roots of quadratic equations.

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  45. L

    Is a Quadratic Equation with b=0 or c=0 Still a Quadratic?

    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...
  46. T

    GEBRA: How to Create Quadratic Equations for a Given Area of a Rectangle

    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...
  47. T

    What Determines the Domain for Inverses of Quadratic Equations?

    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...