Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "One Root Equations Document"

m (→‎1.1 Deriving the quadratic formula)
m (removed unnecessary crediting of contributors)
 
(11 intermediate revisions by 2 users not shown)
Line 1: Line 1:
−
'''One root equations'''
 
−
'''Written by Justin Stevens '''
 
−
 
−
 
−
 
 
==  Deriving the quadratic formula ==
 
==  Deriving the quadratic formula ==
  
Line 40: Line 35:
 
Let’s evaluate cases now.
 
Let’s evaluate cases now.
  
−
==1.2 The discriminant when a= c ==
+
==  The discriminant when a= c ==
  
 
First we evaluate the case <math>a=c</math>
 
First we evaluate the case <math>a=c</math>
Line 56: Line 51:
 
Therefore when <math>a=c</math> and <math>b=2a</math> we are going to have one root for the equation.
 
Therefore when <math>a=c</math> and <math>b=2a</math> we are going to have one root for the equation.
  
−
=== Example 1: ===
+
=== Example 1 ===
  
 
<math>9x^2+18x+9x=0</math>
 
<math>9x^2+18x+9x=0</math>
Line 69: Line 64:
  
  
−
=== Example 2: ===
+
=== Example 2 ===
  
 
<math>x^2+2x+1=0</math>
 
<math>x^2+2x+1=0</math>
Line 112: Line 107:
 
'''So, when <math>a=c=\frac{b}{2}</math> we will have one root, and that one root will be –1.'''
 
'''So, when <math>a=c=\frac{b}{2}</math> we will have one root, and that one root will be –1.'''
  
−
== 1.3 The discriminant when a=b ==
+
==  The discriminant when a=b ==
  
 
When <math>a=b</math> we get:
 
When <math>a=b</math> we get:
Line 130: Line 125:
 
So, our only case is <math>a=b=4c</math>
 
So, our only case is <math>a=b=4c</math>
  
−
=== Example 1:  ===
+
=== Example 1 ===
 
<math>4x^2+4x+1=0</math>
 
<math>4x^2+4x+1=0</math>
  
Line 139: Line 134:
 
<math>\frac{-4}{8}=\boxed{\frac{-1}{2}}</math>
 
<math>\frac{-4}{8}=\boxed{\frac{-1}{2}}</math>
  
−
=== Example 2:  ===
+
=== Example 2 ===
 
<math>8x^2+8x+2=0</math>
 
<math>8x^2+8x+2=0</math>
  
Line 173: Line 168:
 
<math>x=\boxed{\frac{-1}{2}}</math>  
 
<math>x=\boxed{\frac{-1}{2}}</math>  
  
−
'''Therefore when <math>a=b=4c</math> we will have one root which will be <math>\frac{-1}{2}</math>'''  
+
'''Therefore when <math>a=b=4c</math> we will have one root which will be <math>\frac{-1}{2}</math>'''
  
−
== 1.4  The discriminant when b=c ==
+
==   The discriminant when b=c ==
  
 
<math>b^2-4ac=0</math>
 
<math>b^2-4ac=0</math>
Line 202: Line 197:
  
  
−
=== Example 1: ===
+
=== Example 1 ===
 
<math>x^2+4x+4=0</math>
 
<math>x^2+4x+4=0</math>
  
Line 209: Line 204:
 
<math>x=-2</math>
 
<math>x=-2</math>
  
−
=== Example 2: ===
+
=== Example 2 ===
 
<math>2x^2+8x+8=0</math>
 
<math>2x^2+8x+8=0</math>
  
Line 236: Line 231:
 
'''Therefore when <math>b=c=4a</math> in <math>ax^2+bx+c=0</math>, the root is going to be <math>-2</math>'''
 
'''Therefore when <math>b=c=4a</math> in <math>ax^2+bx+c=0</math>, the root is going to be <math>-2</math>'''
  
 +
==  Conclusion ==
 +
When <math>a=\frac{b}{2}=c</math> we will have one root.  That one root will be –1.
  
 +
When <math>a=b=4c</math> we will have one root.  That one root will be <math>\frac{-1}{2}</math>.
  
−
== 1.5 Conclusion ==
+
When <math>4a=b=c</math> we will have one root.  That one root will be –2.
−
When <math>a=\frac{b}{2}=c</math> we will have one root.  That one root will be –1.
 
−
When <math>a=b=4c</math> we will have one root.  That one root will be <math>\frac{-1}{2}</math>
 
−
When <math>4a=b=c</math> we will have one root.  That one root will be –2
 
−
 
 
−
== Contributors ==
 
−
<nowiki>Thanks to our contributors:
 
−
AIME15
 
−
El_Ectric
 
−
BOGTRO
 
−
bzprules
 
−
El_Ectric
 
−
SuperNerd123
 
−
</nowiki>
 

Latest revision as of 18:50, 2 June 2011

Deriving the quadratic formula

When we have a quadratic, we normally specify it in the form $ax^2+bx+c=0$

The way to find a formula for the roots of the equations goes as follows:

\[ax^2+bx+c=0\] \[\frac{ax^2+bx+c}{a}=\frac{0}{a}\] \[x^2+\frac{bx}{a}+\frac{c}{a}=0\] \[\text{We complete the square to get}\] \[(x+\frac{b}{2a})^2+\frac{c}{a}=\frac{b^2}{4a^2}\] \[(x+\frac{b}{2a})^2=\frac{b^2}{4a^2}-\frac{c}{a}\] \[(x+\frac{b}{2a})^2=\frac{b^2}{4a^2}-\frac{c}{a}*\frac{4a}{4a}\] \[(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\] 

\[\sqrt{(x+\frac{b}{2a})^2=\frac{b^2-4ac}{4a^2}\] (Error compiling LaTeX. Unknown error_msg)

\[x+\frac{b}{2a}=\pm \frac{\sqrt{b^2-4ac}}{2a}\] \[x+\frac{b}{2a}-\frac{b}{2a}=\pm \frac{\sqrt{b^2-4ac}}{2a}-\frac{b}{2a}\] \[x=\frac{-b \pm \sqrt{b^2-4ac}}{2a}\]

We have two real roots when $\sqrt{b^2-4ac}> 0$ We have complex roots when $\sqrt{b^2-4ac}<0$ We have one root when $\sqrt{b^2-4ac}=0$

We are going to analyze the case when $\sqrt{b^2-4ac}=0$

$b^2-4ac$ is called the discriminant.

Square both sides to get: $b^2-4ac=0$ Add $4ac$ to both sides to get:

$b^2=4ac$


Let’s evaluate cases now.

The discriminant when a= c

First we evaluate the case $a=c$

When a=c, we have:

$b^2=4a^2$

$b^2=4c^2$

Take the square root of both sides to get: $b=2c$ $b=2a$

Therefore when $a=c$ and $b=2a$ we are going to have one root for the equation.

Example 1

$9x^2+18x+9x=0$

$\frac{-18 \pm \sqrt{(18)^2-4*9*9}}{2*9}$

$\frac{-18 \pm \sqrt{(9^2*2^2)-(9^2*2^2)}}{2*9}$

$\frac{-18 \pm \sqrt{0}}{18}$

$\frac{-18}{18}=\boxed{-1}$


Example 2

$x^2+2x+1=0$

$\frac{-2 \pm \sqrt{2^2-4*1*1}}{2*1}$

$\frac{-2 \pm \sqrt{4-4}}{2}$

$\frac{-2 \pm \sqrt{0}}{2}$

$\frac{-2}{2}=\boxed{-1}$

When $a=c$ and $b=2a$ we can write the problem as follows:

$ax^2+2ax+a=0$

Using the quadratic formula, we get: $\frac{-2a \pm \sqrt{(2a)^2-4*a*a}}{2*a}$

$\frac{-2a \pm \sqrt{4a^2-4a^2}}{2a}$

$\frac{-2a \pm \sqrt{0}}{2a}$

$\frac{-2a}{2a}=\boxed{-1}$

A simpler way to do this, is to start off by dividing by a. We get: $x^2+2x+1=0$ $(x+1)^2=0$ $x=\boxed{-1}$

If we want to write this as one equality, we can do as follows:

$a=c$ $b=2a$

Divide by 2 on the second equation to get:

$\frac{b}{2}=a$

Therefore we have $a=c=\frac{b}{2}$

So, when $a=c=\frac{b}{2}$ we will have one root, and that one root will be –1.

The discriminant when a=b

When $a=b$ we get:

$b^2-4ac=0$ $a^2-4ac=0$ $a(a-4c)=0$

The roots are going to be $a=0, a=4c$

Therefore we have: $a=b=0$ or $a=b=4c$

However, if (a,b)=0, we would be left with $(0*x^2+0x+c)=0$ This is only a constant term left.

So, our only case is $a=b=4c$

Example 1

$4x^2+4x+1=0$

$\frac{-4 \pm \sqrt{4^2-4*4*1}}{2*4}$

$\frac{-4 \pm \sqrt{16-16}}{8}$

$\frac{-4}{8}=\boxed{\frac{-1}{2}}$

Example 2

$8x^2+8x+2=0$

$\frac{-8 \pm \sqrt{8^2-4*8*2}}{2*8}$

$\frac{-8 \pm \sqrt{64-64}}{16}$

$\frac{-8}{16}=\boxed{\frac{-1}2}$

Let’s find a general formula for this. If $a=b=4c$, we have:

$4cx^2+4cx+c=0$

Using the quadratic formula, we get: $\frac{-4c \pm \sqrt{4c^2-4*4c*c}}{2*4c}$

$\frac{-4c \pm \sqrt{4c^2-4c^2}}{8c}$

$\frac{-4c \pm 0}{8c}$

$\frac{-4c}{8c}$

$\frac{-4*\cancel{c}}{8*\cancel{c}}=\boxed{\frac{-1}2}$

Another way to do this, is to start off by dividing by c. $4x^2+4x+1=0$ Divide by 4 to get: $x^2+x+\frac{1}{4}$

$(x+\frac{1}{2})^2=0$

$x=\boxed{\frac{-1}{2}}$

Therefore when $a=b=4c$ we will have one root which will be $\frac{-1}{2}$

The discriminant when b=c

$b^2-4ac=0$ $b=c$, therefore $c^2-4ac=0$ $c(c-4a)=0$

The roots are going to be $c=0$ which gives us: $0(0-4a)=0$

Since $b=c$ we get:

$ax^2+0*x+0=0$ $ax^2=0$

Divide by a to get: $x^2=0$ $x=0$

However, since (a,b,x)=(0,0,0) this is just a case with only a quadratic term, and shouldn’t be considered.

Let’s try the second case of $c(c-4a)$ which is going to be $c=4a$

We have: $b=c=4a$


Example 1

$x^2+4x+4=0$

$(x+2)^2=0$

$x=-2$

Example 2

$2x^2+8x+8=0$

$\frac{-8 \pm \sqrt{8^2-4*8*2}}{2*2}$

$\frac{-8 \pm 0}{4}$

$\frac{-8}{4}=-2$

Let’s try an in-general example of when $b=c=4a$

$ax^2+4ax+4a=0$

Using the quadratic formula, we get $x=\frac{-4a \pm \sqrt{(4a)^2-4*4a*a}}{2*a}$

$x=\frac{-4a \pm \sqrt{16a^2-16a^2}}{2a}$

$x=\frac{-4a}{2a}=-2$

We can also start off by dividing by a to get: $x^2+4x+4=0$ $(x+2)^2=0$ $x=\boxed{-2}$

Therefore when $b=c=4a$ in $ax^2+bx+c=0$, the root is going to be $-2$

Conclusion

When $a=\frac{b}{2}=c$ we will have one root. That one root will be –1.

When $a=b=4c$ we will have one root. That one root will be $\frac{-1}{2}$.

When $4a=b=c$ we will have one root. That one root will be –2.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=One_Root_Equations_Document&oldid=38975"