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Certainly! Let's solve the quadratic expression [tex]\( b^2 - 11b + 18 \)[/tex] step-by-step.
1. Understand the quadratic expression:
The given quadratic expression is [tex]\( b^2 - 11b + 18 \)[/tex].
2. Factor the quadratic expression:
To factor the quadratic expression, we need to find two numbers that multiply to the constant term (18) and add up to the coefficient of the linear term (-11).
Let's find such numbers:
- The constant term is 18.
- The coefficient of the linear term is -11.
We need to find two numbers, say [tex]\( m \)[/tex] and [tex]\( n \)[/tex], such that:
[tex]\( m \times n = 18 \)[/tex]
[tex]\( m + n = -11 \)[/tex]
After examining factor pairs of 18, we observe that (-2) and (-9) satisfy these conditions:
- (-2) \times (-9) = 18
- (-2) + (-9) = -11
Therefore, we can factor the quadratic expression as:
[tex]\[ b^2 - 11b + 18 = (b - 2)(b - 9) \][/tex]
3. Find the solutions by setting the factored expression to zero:
Now, we set the factored expression equal to zero to find the solutions:
[tex]\[ (b - 2)(b - 9) = 0 \][/tex]
According to the zero-product property, if a product of two factors is zero, then at least one of the factors must be zero. So we set each factor to zero:
[tex]\[ b - 2 = 0 \quad \text{or} \quad b - 9 = 0 \][/tex]
Solving these equations:
- For [tex]\( b - 2 = 0 \)[/tex]:
[tex]\[ b = 2 \][/tex]
- For [tex]\( b - 9 = 0 \)[/tex]:
[tex]\[ b = 9 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( b^2 - 11b + 18 = 0 \)[/tex] are:
[tex]\[ b = 2 \quad \text{and} \quad b = 9 \][/tex]
4. Summarize the results:
The factored form of the quadratic expression [tex]\( b^2 - 11b + 18 \)[/tex] is:
[tex]\[ (b - 2)(b - 9) \][/tex]
The solutions to the equation [tex]\( b^2 - 11b + 18 = 0 \)[/tex] are:
[tex]\[ b = 2 \quad \text{and} \quad b = 9 \][/tex]
So the detailed solution shows that the factored form is [tex]\((b - 2)(b - 9)\)[/tex] and the solutions are [tex]\(b = 2\)[/tex] and [tex]\(b = 9\)[/tex].
1. Understand the quadratic expression:
The given quadratic expression is [tex]\( b^2 - 11b + 18 \)[/tex].
2. Factor the quadratic expression:
To factor the quadratic expression, we need to find two numbers that multiply to the constant term (18) and add up to the coefficient of the linear term (-11).
Let's find such numbers:
- The constant term is 18.
- The coefficient of the linear term is -11.
We need to find two numbers, say [tex]\( m \)[/tex] and [tex]\( n \)[/tex], such that:
[tex]\( m \times n = 18 \)[/tex]
[tex]\( m + n = -11 \)[/tex]
After examining factor pairs of 18, we observe that (-2) and (-9) satisfy these conditions:
- (-2) \times (-9) = 18
- (-2) + (-9) = -11
Therefore, we can factor the quadratic expression as:
[tex]\[ b^2 - 11b + 18 = (b - 2)(b - 9) \][/tex]
3. Find the solutions by setting the factored expression to zero:
Now, we set the factored expression equal to zero to find the solutions:
[tex]\[ (b - 2)(b - 9) = 0 \][/tex]
According to the zero-product property, if a product of two factors is zero, then at least one of the factors must be zero. So we set each factor to zero:
[tex]\[ b - 2 = 0 \quad \text{or} \quad b - 9 = 0 \][/tex]
Solving these equations:
- For [tex]\( b - 2 = 0 \)[/tex]:
[tex]\[ b = 2 \][/tex]
- For [tex]\( b - 9 = 0 \)[/tex]:
[tex]\[ b = 9 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( b^2 - 11b + 18 = 0 \)[/tex] are:
[tex]\[ b = 2 \quad \text{and} \quad b = 9 \][/tex]
4. Summarize the results:
The factored form of the quadratic expression [tex]\( b^2 - 11b + 18 \)[/tex] is:
[tex]\[ (b - 2)(b - 9) \][/tex]
The solutions to the equation [tex]\( b^2 - 11b + 18 = 0 \)[/tex] are:
[tex]\[ b = 2 \quad \text{and} \quad b = 9 \][/tex]
So the detailed solution shows that the factored form is [tex]\((b - 2)(b - 9)\)[/tex] and the solutions are [tex]\(b = 2\)[/tex] and [tex]\(b = 9\)[/tex].
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