To solve the quadratic equation [tex]\( x^2 - 12x + 36 = 0 \)[/tex], let's work through it step by step using the method of factoring.
### Step 1: Identify the quadratic equation
The given quadratic equation is:
[tex]\[ x^2 - 12x + 36 = 0 \][/tex]
### Step 2: Factor the quadratic equation
To factor the quadratic equation, we need to express it in the form:
[tex]\[ (x - a)(x - b) = 0 \][/tex]
where [tex]\( a \)[/tex] and [tex]\( b \)[/tex] are numbers such that:
[tex]\[ a + b = 12 \][/tex]
[tex]\[ ab = 36 \][/tex]
We look for two numbers that add up to 12 and multiply to 36.
By inspection or trial and error, we can see that:
[tex]\[ 6 + 6 = 12 \][/tex]
[tex]\[ 6 \times 6 = 36 \][/tex]
So, we can factor the quadratic equation as:
[tex]\[ (x - 6)(x - 6) = 0 \][/tex]
### Step 3: Solve for [tex]\( x \)[/tex]
Set each factor equal to zero:
[tex]\[ x - 6 = 0 \][/tex]
Solving this gives:
[tex]\[ x = 6 \][/tex]
### Step 4: Verify the solution
To ensure our factored form is correct, we can expand [tex]\( (x - 6)^2 \)[/tex] to see if it matches the original equation.
[tex]\[ (x - 6)(x - 6) = x^2 - 6x - 6x + 36 = x^2 - 12x + 36 \][/tex]
Indeed, it matches the original quadratic equation.
Therefore, the solution to the quadratic equation [tex]\( x^2 - 12x + 36 = 0 \)[/tex] is:
[tex]\[ x = 6 \][/tex]
