Equation and its Roots
p.1. Definition of Equation and its Root
An equation in one variable x is the equality f(x)=g(x), with the task to find all values of the variable x that make this equality a numerical identity.
The value of the variable x, for which the expressions f(x) and g(x) take equal numerical values, is called a root of the equation f(x)=g(x).
For example, the equation 15x+8=23 has one root x=1.
The equation x(x + 5)(x - 3) = 0 has three roots, $x_1 = 0,x_2 = -5,x_3 = 3$.
The equation $x^2 = -1$ has no real roots.
The equation 5(x + 3)=5x + 15 has infinitely many roots, since it is a true equality for any $x \in \Bbb R$, an identity.
Solving an equation means finding all of its roots or proving that there are none.
p.2. Examples of Problem Solving
Example 1. Solve the equation and check the result x - (3 - 2x) = 9
Solution:
x-(3-2x)=9 $\iff$ x-3+2x=9 $\iff$ x+2x=9+3 $\iff$ 3x=12 $\iff$ x=4
Check:
$4 -(3 - 2 \cdot 4)=9 \implies 4 - 3 + 8 = 9 \implies 9 \equiv 9$
Answer: x = 4
Example 2. Solve the equation and check the result 7(x + 3) = 56
Solution:
7(x + 3)=56 |:7 $\iff$ x + 3 = 8 $\iff$ x = 8 - 3 $\iff$ x=5
Check:
$7(5 + 3) = 56 \implies 7 \cdot 8 = 56 \implies 56 \equiv 56$
Answer: x = 5
Example 3. Solve the equation and check the result (3x + 4) : 2 = 14
Solution:
(3x + 4) : 2=14 |$\times$2 $\iff$ 3x + 4 = 28 $\iff$ 3x = 28 - 4 $\iff$ 3x = 24 $\iff$ x=8
Check:
$(3 \cdot 8 + 4) : 2 = 14 \implies (24 + 4) : 2 = 14 \implies 28 : 2 = 14 \implies 14 \equiv 14$
Answer: x = 8
Example 4. Solve the equation $\frac {2x - 7}{2} = \frac {3x+6}{3}$
Solution:
$\frac {2x-7}{2}=\frac {x+6}{3} | \times 6 \iff 3(2x-7)=2(x+6) \iff 6x-21=2x+12 \iff $
$\iff 6x-2x=12+21 \iff 4x=33 \iff x= \frac {33}{4} =8 \frac 14$
Answer: $8 \frac 14$
Example 5. Solve the equation |x+1|=5
Solution:
$$|x+1|=5 \iff \left[ \begin{array}{cc} {x+1=-5}\\ {x+1=5} \end{array} \right. \iff \left[ \begin{array}{cc} {x=-5-1}\\ {x=5-1} \end{array} \right. \iff \left[ \begin{array}{cc} {x_1=-6}\\ {x_2=4} \end{array} \right. $$
Answer: $ x_1=-6, x_2=4$
Example 6*. Solve the equation and check the result |x + 1| = x + 3
Solution:
$$ |x + 1| = x + 3 \iff \left[ \begin{array}{cc} {\left\{ \begin{array}{c} x+1 \ge 0 \\ x+1=x+3 \end{array} \right.}\\ {\left\{ \begin{array}{c} x+1<0 \\ -(x+1)=x+3 \end{array} \right.} \end{array} \right. \iff \left[ \begin{array}{cc} {\left\{ \begin{array}{c} x \ge -1 \\ 1=3 \end{array} \right.}\\ {\left\{ \begin{array}{c} x<-1 \\ -x-1=x+3 \end{array} \right.} \end{array} \right. \iff $$
$$ \iff \left[ \begin{array}{cc} {\emptyset}\\ {\left\{ \begin{array}{c} x<-1 \\ -x-x=3+1 \end{array} \right.} \end{array} \right. \iff \left[ \begin{array}{cc} {x<-1}\\ {-2x=4} \end{array} \right. \iff \left[ \begin{array}{cc} {x<-1}\\ {x=-2} \end{array} \right. \iff x=-2 $$
Check:
$$|-2+1|=-2+3 \implies |-1|=1\implies 1 \equiv 1$$
Answer: x = -2
Example 7. Find the value of the parameter a such that the equation 5ax+18=3 has the root x=-3?
Solution:
Substitute x=-3 into the equation and solve it for the variable a:
5a $\cdot$ (-3) + 18 = 3 $\iff$ -15a = 3 - 18 $\iff$ -15a = -15 $\iff$ a = -15:(-15)=1
a=1
Answer: a = 1
