16 Exercise C5: QUADRATIC 3-TERM (Gr10) Factorize completely 1. \( 2 x^{2}-7 x-72 \) 2. \( 2 x^{2}-5 x-12 \) 3. \( 3 x^{2}-13 x-10 \) 4. \( 2 x^{2}-9 x+9 \) 5. \( 3 x^{2}+20 x+12 \) 6. \( 2 x^{2}+7 x-15 \) 7. \( 3 x^{2}-4 x-15 \) 8. \( 7 x^{2}-10 x-8 \) 9. \( 11 x^{2}-25 x+6 \) 10. \( 15 x^{2}-7 x-2 \)

Sure! Let's factorize each of the given quadratic expressions completely. ### 1. \( 2x^{2} - 7x - 72 \) **Factorization:** \[ 2x^{2} - 7x - 72 = (2x + 9)(x - 8) \] ### 2. \( 2x^{2} - 5x - 12 \) **Factorization:** \[ 2x^{2} - 5x - 12 = (2x + 3)(x - 4) \] ### 3. \( 3x^{2} - 13x - 10 \) **Factorization:** \[ 3x^{2} - 13x - 10 = (3x + 2)(x - 5) \] ### 4. \( 2x^{2} - 9x + 9 \) **Factorization:** \[ 2x^{2} - 9x + 9 = (2x - 3)(x - 3) \] ### 5. \( 3x^{2} + 20x + 12 \) **Factorization:** \[ 3x^{2} + 20x + 12 = (3x + 2)(x + 6) \] ### 6. \( 2x^{2} + 7x - 15 \) **Factorization:** \[ 2x^{2} + 7x - 15 = (2x - 3)(x + 5) \] ### 7. \( 3x^{2} - 4x - 15 \) **Factorization:** \[ 3x^{2} - 4x - 15 = (3x + 5)(x - 3) \] ### 8. \( 7x^{2} - 10x - 8 \) **Factorization:** \[ 7x^{2} - 10x - 8 = (7x + 4)(x - 2) \] ### 9. \( 11x^{2} - 25x + 6 \) **Factorization:** \[ 11x^{2} - 25x + 6 = (11x - 2)(x - 3) \] ### 10. \( 15x^{2} - 7x - 2 \) **Factorization:** \[ 15x^{2} - 7x - 2 = (5x + 1)(3x - 2) \] --- ### **Methodology for Factorization** For each quadratic expression of the form \( ax^{2} + bx + c \), follow these steps: 1. **Multiply \( a \) and \( c \):** Compute \( ac \). 2. **Find Two Numbers:** Identify two numbers \( m \) and \( n \) such that: - \( m \times n = ac \) - \( m + n = b \) 3. **Rewrite the Middle Term:** Express \( bx \) as \( mx + nx \). 4. **Factor by Grouping:** Group the terms and factor out the common factors. 5. **Write the Final Factors:** Combine the grouped factors to get the final factored form. **Example:** Let's take the first quadratic expression as an example. **Given:** \[ 2x^{2} - 7x - 72 \] 1. **Multiply \( a \) and \( c \):** \[ 2 \times (-72) = -144 \] 2. **Find Two Numbers:** We need two numbers that multiply to \(-144\) and add up to \(-7\). These numbers are \( 9 \) and \(-16\) because: \[ 9 \times (-16) = -144 \quad \text{and} \quad 9 + (-16) = -7 \] 3. **Rewrite the Middle Term:** \[ 2x^{2} + 9x - 16x - 72 \] 4. **Factor by Grouping:** \[ (2x^{2} + 9x) + (-16x - 72) = x(2x + 9) - 8(2x + 9) \] 5. **Write the Final Factors:** \[ (2x + 9)(x - 8) \] Repeat these steps for each quadratic expression to achieve complete factorization.