Answer :
Sure! Let's factor the quadratic equation \(x^2 - 13x + 12\) step-by-step.
### Step 1: Write down the quadratic equation
We start with the quadratic equation:
[tex]\[x^2 - 13x + 12\][/tex]
### Step 2: Identify the coefficients
For any quadratic equation of the form \(ax^2 + bx + c\), the coefficients are:
- \(a = 1\)
- \(b = -13\)
- \(c = 12\)
### Step 3: Find factors of the constant term
We need to find two numbers that multiply to give \(c = 12\) and add up to give \(b = -13\).
The possible pairs of factors of 12 are:
- \(1 \cdot 12\)
- \(2 \cdot 6\)
- \(3 \cdot 4\)
To find the correct pair, we also need to take their signs into account to add up to \(-13\):
1. The pair \((1, 12)\):
[tex]\[ 1 + 12 = 13 \][/tex]
[tex]\[ 1 \cdot 12 = 12 \][/tex]
Thus, the pair does not work since their sum is positive 13, not negative 13.
2. The pair \((2, 6)\):
[tex]\[ 2 + 6 = 8 \][/tex]
[tex]\[ 2 \cdot 6 = 12 \][/tex]
Thus, the pair does not work since their sum is 8.
3. The pair \((3, 4)\):
[tex]\[ 3 + 4 = 7 \][/tex]
[tex]\[ 3 \cdot 4 = 12 \][/tex]
Thus, the pair does not work since their sum is 7.
Considering we seek factors that sum to \(-13\), appropriate negative pairs should be:
- \((-1) \cdot (-12)\)
- \((-2) \cdot (-6)\)
- \((-3) \cdot (-4)\)
4. Only considering \((-1, -12)\):
[tex]\[ -1 + (-12) = -13 \][/tex]
[tex]\[ -1 \cdot (-12) = 12 \][/tex]
This pair works since the sum is \(-13\).
### Step 4: Write the factorized form
Given that the valid pair of numbers is \(-1\) and \(-12\), we can now express the original quadratic equation in its factored form:
[tex]\[ (x - 12)(x - 1) \][/tex]
### Conclusion
Thus, the quadratic equation \(x^2 - 13x + 12\) factors over the integers as:
[tex]\[ (x - 12)(x - 1) \][/tex]
### Step 1: Write down the quadratic equation
We start with the quadratic equation:
[tex]\[x^2 - 13x + 12\][/tex]
### Step 2: Identify the coefficients
For any quadratic equation of the form \(ax^2 + bx + c\), the coefficients are:
- \(a = 1\)
- \(b = -13\)
- \(c = 12\)
### Step 3: Find factors of the constant term
We need to find two numbers that multiply to give \(c = 12\) and add up to give \(b = -13\).
The possible pairs of factors of 12 are:
- \(1 \cdot 12\)
- \(2 \cdot 6\)
- \(3 \cdot 4\)
To find the correct pair, we also need to take their signs into account to add up to \(-13\):
1. The pair \((1, 12)\):
[tex]\[ 1 + 12 = 13 \][/tex]
[tex]\[ 1 \cdot 12 = 12 \][/tex]
Thus, the pair does not work since their sum is positive 13, not negative 13.
2. The pair \((2, 6)\):
[tex]\[ 2 + 6 = 8 \][/tex]
[tex]\[ 2 \cdot 6 = 12 \][/tex]
Thus, the pair does not work since their sum is 8.
3. The pair \((3, 4)\):
[tex]\[ 3 + 4 = 7 \][/tex]
[tex]\[ 3 \cdot 4 = 12 \][/tex]
Thus, the pair does not work since their sum is 7.
Considering we seek factors that sum to \(-13\), appropriate negative pairs should be:
- \((-1) \cdot (-12)\)
- \((-2) \cdot (-6)\)
- \((-3) \cdot (-4)\)
4. Only considering \((-1, -12)\):
[tex]\[ -1 + (-12) = -13 \][/tex]
[tex]\[ -1 \cdot (-12) = 12 \][/tex]
This pair works since the sum is \(-13\).
### Step 4: Write the factorized form
Given that the valid pair of numbers is \(-1\) and \(-12\), we can now express the original quadratic equation in its factored form:
[tex]\[ (x - 12)(x - 1) \][/tex]
### Conclusion
Thus, the quadratic equation \(x^2 - 13x + 12\) factors over the integers as:
[tex]\[ (x - 12)(x - 1) \][/tex]
