What is the answer to the expression \(\frac{x+7}{7x+35} \cdot \frac{x^2-3x-40}{x-8}\)?
Answer (2)
7 x + 35 x + 7 ⋅ x − 8 x 2 − 3 x − 40 = ( ∗ ) x 2 − 3 x − 40 = 0 a = 1 ; b = − 3 ; c = − 40 Δ = b 2 − 4 a c ; x 1 = 2 a − b − Δ ; x 2 = 2 a − b + Δ Δ = ( − 3 ) 2 − 4 ⋅ 1 ⋅ ( − 40 ) = 9 + 160 = 169 ; Δ = 169 = 13 x 1 = 2 ⋅ 1 3 − 13 = 2 − 10 = − 5 ; x 2 = 2 ⋅ 1 3 + 13 = 2 16 = 8 x 2 − 3 x − 40 = ( x + 5 ) ( x − 8 ) ( ∗ ) = 7 ( x + 5 ) x + 7 ⋅ x − 8 ( x + 5 ) ( x − 8 ) = 7 x + 7 = 7 x + 7 7 = 7 1 x + 1 D : x = − 5 ∧ x = 8
The expression 7 x + 35 x + 7 ⋅ x − 8 x 2 − 3 x − 40 simplifies to 7 1 x + 1 after factoring and canceling common terms. The values x = − 5 and x = 8 are not allowed since they make the denominator zero. Thus, the simplified form is valid for all other values of x .
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