How to solve the equation:
\[ 4n^2 - 3n - 7 = 0 \]
Answer (3)
4 n 2 − 3 n − 7 = 0 ( ∗ ) ( 2 n ) 2 − 2 ⋅ 2 n ⋅ 4 3 + ( 4 3 ) 2 − ( 4 3 ) 2 = 7 ( 2 n − 4 3 ) 2 − 16 9 = 7 ( 2 n − 4 3 ) 2 = 7 + 16 9 ( 2 n − 4 3 ) 2 = 16 112 + 16 9 ( 2 n − 4 3 ) 2 = 16 121 → 2 n − 4 3 = 16 121 ∨ 2 n − 4 3 = − 16 121
2 n = 4 11 + 4 3 ∨ 2 n = − 4 11 + 4 3 2 n = 4 14 ∨ 2 n = − 4 8 2 n = 2 7 ∨ 2 n = − 2 = 4 7 ∨ n = − 1
( a − b ) 2 = a 2 − 2 ab + b 2
4 n 2 − 3 n − 7 = 0 a = 4 , b = − 3 c = − 7 Δ = b 2 − 4 a c = ( − 3 ) 2 − 4 ∗ 4 ∗ ( − 7 ) = 9 + 112 = 121
x 1 = 2 a − b − Δ = 2 ∗ 4 3 − 121 = 8 3 − 11 = 8 − 8 = − 1 x 2 = 2 a − b + Δ = 2 ∗ 4 3 + 121 = 8 3 + 11 = 8 14 = 4 7 = 1 4 3
The solutions to the quadratic equation 4 n 2 − 3 n − 7 = 0 are n = 4 7 and n = − 1 , found using the quadratic formula. By substituting the coefficients into the formula, we calculated the roots step-by-step. This method ensures we accurately find the values of n that satisfy the equation.
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