5- \( \frac{d^{2}-9}{d^{2}-7 d+12} \)

Simplify the expression by following steps: - step0: Solution: \(5-\frac{\left(d^{2}-9\right)}{\left(d^{2}-7d+12\right)}\) - step1: Remove the parentheses: \(5-\frac{d^{2}-9}{d^{2}-7d+12}\) - step2: Divide the terms: \(5-\frac{d+3}{d-4}\) - step3: Reduce fractions to a common denominator: \(\frac{5\left(d-4\right)}{d-4}-\frac{d+3}{d-4}\) - step4: Transform the expression: \(\frac{5\left(d-4\right)-\left(d+3\right)}{d-4}\) - step5: Multiply the terms: \(\frac{5d-20-\left(d+3\right)}{d-4}\) - step6: Subtract the terms: \(\frac{4d-23}{d-4}\) To simplify the expression \( 5 - \frac{d^{2}-9}{d^{2}-7d+12} \), we can follow these steps: 1. **Identify the components**: - The expression consists of a constant \( 5 \) and a fraction \( \frac{d^{2}-9}{d^{2}-7d+12} \). 2. **Factor the denominator**: - The denominator \( d^{2}-7d+12 \) can be factored. It factors to \( (d-3)(d-4) \). 3. **Rewrite the expression**: - The expression can be rewritten as: \[ 5 - \frac{(d-3)(d+3)}{(d-3)(d-4)} \] 4. **Simplify the fraction**: - The \( (d-3) \) terms cancel out (assuming \( d \neq 3 \)): \[ 5 - \frac{d+3}{d-4} \] 5. **Combine the terms**: - To combine \( 5 \) and \( -\frac{d+3}{d-4} \), we express \( 5 \) with a common denominator: \[ 5 = \frac{5(d-4)}{d-4} = \frac{5d - 20}{d-4} \] - Now, we can combine: \[ \frac{5d - 20}{d-4} - \frac{d+3}{d-4} = \frac{(5d - 20) - (d + 3)}{d-4} \] 6. **Simplify the numerator**: - Combine the terms in the numerator: \[ 5d - 20 - d - 3 = 4d - 23 \] 7. **Final expression**: - Thus, the simplified expression is: \[ \frac{4d - 23}{d - 4} \] So, the final result is: \[ \frac{4d - 23}{d - 4} \]