The Quadratic Formula

Key Questions

  • What is the quadratic formula used for?

    The quadratic formula is used to get the roots of a quadratic equation, if the roots exists at all.

    We usually just perform factorization to get the roots of a quadratic equation. However, this is not always possible (especially when the roots are irrational)

    The quadratic formula is

    x=−b±2√b2−4ac2a

    Example 1:

    y=x2−3x−4
    0=x2−3x−4

    ⇒0=(x−4)(x+1)
    ⇒x=4,x=−1

    Using the quadratic formula, let's try to solve the same equation

    x=−(−3)±2√(−3)2−4⋅1⋅(−4)2⋅1
    ⇒x=3±2√9+162
    ⇒x=3±2√252
    ⇒x=3+52,x=3−52
    ⇒x=4,x=−1

    Example 2:

    y=2x2−3x−5
    0=2x2−3x−5

    Performing factorization is a little hard for this equation, so let's jump straight to using the quadratic formula

    x=−(−3)±2√(−3)2−4⋅2⋅(−5)2⋅2

    x=3±2√9+404

    x=3±2√494

    x=3+74,x=3−74

    x=52,x=−1

    seph · · Oct 19 2014
  • What is an example of using the quadratic formula?

    Suppose that you have a function represented by f(x)=Ax2+Bx+C.

    We can use the quadratic formula to find the zeroes of this function, by setting f(x)=Ax2+Bx+C=0.

    Technically we can also find complex roots for it, but typically one will be asked to work only with real roots. The quadratic formula is represented as:

    −B±√B2−4AC2A=x

    ... where x represents the x-coordinate of the zero.

    If B2−4AC<0, we will be dealing with complex roots, and if B2−4AC≥0, we will have real roots.

    As an example, consider the function x2−13x+12. Here,

    A=1,B=−13,C=12.

    Then for the quadratic formula we would have:

    x=13±√(−13)2−4(1)(12)2(1) =

    13±√169−482=13±112

    Thus, our roots are x=1 and x=12.

    For an example with complex roots, we have the function f(x)=x2+1. Here A=1,B=0,C=1.

    Then by the quadratic equation,

    x=0±√02−4(1)(1)2(1)=±√−42=±i

    ... where i is the imaginary unit, defined by its property of i2=−1.

    In the graph for this function on the real coordinate plane, we will see no zeroes, but the function will have these two imaginary roots.

    Psykolord1989 . · 1 · Nov 2 2014
  • What is the discriminant and how do you find it?

    The discriminant is part of the quadratic formula.

    Quadratic Formula

    x=−b±√b2−4ac2a

    Discriminant

    b2−4ac

    The discriminant tells you the number and types of solutions to a quadratic equation.

    b2−4ac=0, one real solution

    b2−4ac>0, two real solutions

    b2−4ac<0, two imaginary solutions

    AJ Speller · · Oct 22 2014
  • What is the quadratic formula?

    Answer:

    x=−b±√b2−4ac2a

    Explanation:

    Negative b plus minus the square root of b squared minus 4*a*c over 2*a. To plug something into the quadratic formula the equation needs to be in standard form (ax2+bx2+c).

    hope this helps!

    Lauren · 1 · Mar 26 2018

Questions

Linear and Quadratic Functions