Lesson Who is who in quadratic equations

This Lesson (Who is who in quadratic equations) was created by by ikleyn(53942)

: View Source, Show
About ikleyn:

Who is who in quadratic equations

The lessons PROOF of quadratic formula... and Introduction into quadratic equations of this module explain what is the quadratic formula and how to use it
to solve quadratic equations.
So, I suppose you know these issues.
You will know more and understand it better after reading this lesson.

Let's consider a quadratic equation

a%2Ax%5E2%2Bb%2Ax%2Bc=0

. (1)

The Figure 1 shows the typical plot of the quadratic function, which is
the left side of this equation.

I guess you saw such plots many times.

The curve in the Figure 1 is called a parabola.

Figure 1. A parabola

Let's perform an operation of square completing to get

(2)

For which value of x the quadratic function (1), (2) has the minimum/maximum?
The answer is: for x=-b%2F%282%2Aa%29

.
To check it, simply substitute this value of x into quadratic function (2),
and you will see that the term a%2A%28x%2Bb%2F%282%2Aa%29%29%5E2

is zero for this value of x.
It is also easy to see that vertical line x=-b%2F%282%2Aa%29

is the symmetry line
of the parabola. The Figure 2 illustrates this fact by showing the symmetry line
colored in green.

Figure 2. Location of the minimum
of a quadratic function

Next question is: what is the value of the quadratic function (1), (2) at x=-b%2F%282%2Aa%29

?
In order to answer this question, simply substitute x=-b%2F%282%2Aa%29

into the function (2),
and you will get the value f%28x%29=-%28b%5E2-4%2Aa%2Ac%29%2F%284%2Aa%29

. This value is marked by the blue
horizontal line in the Figure 3.

Figure 3. The minimum value
of a quadratic function

Last question: What is the distance along the X-axis from the green vertical symmetry line
to the root(s) of the quadratic function? The answer is sqrt%28b%5E2-4%2Aa%2Ac%29%2F%282%2Aa%29

, exactly the value produced by the expression of the standard quadratic formula after the sign(s) +/-.

My other lessons on quadratic equations in this website are
    - Introduction into Quadratic Equations
    - PROOF of quadratic formula by completing the square

    - HOW TO complete the square - Learning by examples
    - HOW TO solve quadratic equation by completing the square - Learning by examples
    - Solving quadratic equations without quadratic formula
    - Using Vieta's theorem to solve quadratic equations and related problems

    - Find a number using quadratic equations
    - Find an equation of the parabola passing through given points
    - Problems on quadratic equations to solve them using discriminant
    - Relative position of a straight line and a parabola on a coordinate plane
    - Advanced minimax problems to solve them using the discriminant principle

    - Using quadratic equations to solve word problems
    - Word problems on engineering constructions of parabolic shapes
    - Challenging word problems solved using quadratic equations
    - Business-related problems to solve them using quadratic equations
    - Rare beauty investment problem to solve using quadratic equation
    - HOW TO solve the problem on quadratic equation mentally and avoid boring calculations
    - Entertainment problems on quadratic equations
    - Prime quadratic polynomials with real coefficients

    - Problem on a projectile moving vertically up and down
    - Problem on an arrow shot vertically upward
    - Problem on a ball thrown vertically up from the top of a tower
    - Problem on a toy rocket launched vertically up from a tall platform

    - Problems on the area and the dimensions of a rectangle
    - Problems on the area and the dimensions of a rectangle surrounded by a strip
    - Problems on a circular pool and a walkway around it

    - OVERVIEW of lessons on solving quadratic equations

Use this file/link ALGEBRA-I - YOUR ONLINE TEXTBOOK to navigate over all topics and lessons of the online textbook ALGEBRA-I.

This lesson has been accessed 4997 times.