Lesson Who is who in quadratic equations

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<H2>Who is who in quadratic equations</H2>

The lessons &nbsp;<A HREF=http://www.algebra.com/algebra/homework/quadratic/lessons/proof-of-quadratic-by-completing-the-square.lesson>PROOF of quadratic formula...</A>&nbsp; and &nbsp;<A HREF=http://www.algebra.com/algebra/homework/quadratic/lessons/Introduction-Into-Quadratics.lesson>Introduction into quadratic equations</A>&nbsp; of this module explain what is the quadratic formula and how to use it

to solve quadratic equations.

So, &nbsp;I suppose you know these issues.

You will know more and understand it better after reading this lesson.

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Let's consider a quadratic equation

{{{a*x^2+b*x+c=0}}}. &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1)

The &nbsp;<B>Figure 1</B>&nbsp; shows the typical plot of the quadratic function, &nbsp;which is&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

the left side of this equation.

I guess you saw such plots many times.

The curve in the &nbsp;<B>Figure 1</B>&nbsp; is called a &nbsp;<B>parabola</B>.

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{{{graph(200, 200,
         -1, 4,
         -2, 2,
         x^2-4*x+3
)}}}

&nbsp;&nbsp;&nbsp;&nbsp;<B>Figure 1</B>. &nbsp;A parabola

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Let's perform an operation of &nbsp;<A HREF=http://www.algebra.com/algebra/homework/quadratic/lessons/proof-of-quadratic-by-completing-the-square.lesson>square completing</A>&nbsp; to get

{{{f(x) = a*x^2+b*x+c = a*(x+b/(2*a))^2 - (b^2-4*a*c)/(4*a)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2)

For which value of &nbsp;<B>x</B>&nbsp; the quadratic function &nbsp;(1), &nbsp;(2)&nbsp; has the minimum/maximum?&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

The answer is: &nbsp;for &nbsp;{{{x=-b/(2*a)}}}. 
To check it, &nbsp;simply substitute this value of &nbsp;<B>x</B>&nbsp; into quadratic function &nbsp;(2), 
and you will see that the term &nbsp;{{{a*(x+b/(2*a))^2}}}&nbsp; is zero for this value of &nbsp;<B>x</B>.
It is also easy to see that vertical line &nbsp;{{{x=-b/(2*a)}}}&nbsp; is the symmetry line
of the parabola. &nbsp;The &nbsp;<B>Figure 2</B>&nbsp; illustrates this fact by showing the symmetry line

colored in green.

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{{{graph(200, 200, 
           -1, 4, 
           -2, 2, 
           x^2-4x+3, 
           50*(x-2-0.001)
)}}}
<B>Figure 2</B>. Location of the minimum 
&nbsp;&nbsp;&nbsp;&nbsp;of a quadratic function

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Next question is: &nbsp;what is the value of the quadratic function &nbsp;(1), &nbsp;(2)&nbsp; at &nbsp;{{{x=-b/(2*a)}}}?&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

In order to answer this question, &nbsp;simply substitute &nbsp;{{{x=-b/(2*a)}}}&nbsp; into the function &nbsp;(2),

and you will get the value &nbsp;{{{f(x)=-(b^2-4*a*c)/(4*a)}}}. &nbsp;This value is marked by the blue

horizontal line in the &nbsp;<B>Figure 3</B>.

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{{{graph(200, 200, 
           -1, 4, 
           -2, 2, 
           x^2-4x+3, 
           50*(x-2-0.001),
           0.0001*x-1
)}}}
<B>Figure 3</B>. The minimum value
&nbsp;&nbsp;&nbsp;&nbsp;of a quadratic function

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Last question: &nbsp;What is the distance along the &nbsp;<B>X</B>-axis from the green vertical symmetry line
to the root(s) of the quadratic function? &nbsp;The answer is &nbsp;{{{sqrt(b^2-4*a*c)/(2*a)}}}, &nbsp;exactly the value produced by the expression of the standard quadratic formula after the sign(s) &nbsp;<B>+/-</B>.

My other lessons on quadratic equations in this website are

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Introduction-Into-Quadratics.lesson>Introduction into Quadratic Equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/proof-of-quadratic-by-completing-the-square.lesson>PROOF of quadratic formula by completing the square</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Learning-by-examples-on-HOW-TO-complete-the-square.lesson>HOW TO complete the square - Learning by examples</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/HOW-TO-solve-quadratic-equation-by-completing-the-square-Learning-by-examples.lesson>HOW TO solve quadratic equation by completing the square - Learning by examples</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Solving-quadratic-equations-without-quadratic-formula.lesson>Solving quadratic equations without quadratic formula</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Using-Vieta%27s-theorem-to-solve-qudratic-equations.lesson>Using Vieta's theorem to solve quadratic equations and related problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/Find-a-number-using--quadratic-equations.lesson>Find a number using quadratic equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Find-an-equation-of-the-parabola-passing-through-given-points.lesson>Find an equation of the parabola passing through given points</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Problems-on-quadratic-equations-to-solve-them-using-discriminant.lesson>Problems on quadratic equations to solve them using discriminant</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/Relative-position-of-a-straight-line-and-a-parabola-on-a-coordinate-plane.lesson>Relative position of a straight line and a parabola on a coordinate plane</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Advanced-minimax-problems-to-solve-using-the-discriminant-principle.lesson>Advanced minimax problems to solve them using the discriminant principle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Using-quadratic-equations-to-solve-word-problems.lesson>Using quadratic equations to solve word problems</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Word-problems-on-engineering-constructions-of-parabolic-shapes.lesson>Word problems on engineering constructions of parabolic shapes</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/Challenging-word-problems-solved-using-quadratic-equations.lesson>Challenging word problems solved using quadratic equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Business-related-problems-to-solve-them-using-quadratic-equations.lesson>Business-related problems to solve them using quadratic equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Rare-beaty-investment-problem-to-solve-using-quadratic-equation.lesson>Rare beauty investment problem to solve using quadratic equation</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/How-to-solve-the-problem-on-quadratic-equation-mentally-and-avoid-boring-calculations.lesson>HOW TO solve the problem on quadratic equation mentally and avoid boring calculations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Entertainment-problems-on-quadratic-equations.lesson>Entertainment problems on quadratic equations</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/Prime-quadratic-polynomials-with-real-coefficients.lesson>Prime quadratic polynomials with real coefficients</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-a-projectile-moving-vertically.lesson>Problem on a projectile moving vertically up and down</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-projectile-shooted-vertically-upward.lesson>Problem on an arrow shot vertically upward</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Typical-problems-on-an-projectile-moving-vertically-up-and-down.lesson>Problem on a ball thrown vertically up from the top of a tower</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/word/travel/Problem-on-a-toy-rocket-launched-vertically-up--from-the-top-of-a-platform.lesson>Problem on a toy rocket launched vertically up from a tall platform</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Surface-area/Problems-on-the-area-and-the-perimeter-of-a-rectangle.lesson>Problems on the area and the dimensions of a rectangle</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Surface-area/Problems-on-the-area-of-a-rectangle-surrounded-by-a-strip.lesson>Problems on the area and the dimensions of a rectangle surrounded by a strip</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Surface-area/Circular-pool-and-a-walkway-around.lesson>Problems on a circular pool and a walkway around it</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/OVERVIEW-of-lessons-on-solving-quadratic-equations.lesson>OVERVIEW of lessons on solving quadratic equations</A>

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-I.