I need help factoring the quadratic equation:
\[ 10x^2 + 31x + 24 \]
Answer (3)
10 x 2 + 31 x + 24 a = 10 ; b = 31 ; c = 24 Δ = b 2 − 4 a c Δ = 3 1 2 − 4 ⋅ 10 ⋅ 24 = 961 − 960 = 1 x 1 = 2 a − b − Δ ; x 2 = 2 a − b + Δ Δ = 1 = 1 x 1 = 2 ⋅ 10 − 31 − 1 = 20 − 32 = − 5 8 ; x 2 = 2 ⋅ 10 − 31 + 1 = 20 − 30 = − 2 3 10 x 2 + 31 x + 24 = 10 ( x + 5 8 ) ( x + 2 3 ) = 5 ( x + 5 8 ) ⋅ 2 ( x + 2 3 ) = ( 5 x + 8 ) ( 2 x + 3 )
10 x 2 + 31 x + 24 = = 10 x 2 + 31 x − 15 x + 15 x + 24 = = 10 x 2 + 16 x + 15 x + 24 = = 2 x ( 5 x + 8 ) + 3 ( 5 x + 8 ) = = ( 5 x + 8 ) ( 2 x + 3 )
To factor the quadratic equation 10 x 2 + 31 x + 24 , first find two numbers that multiply to 240 (the product of a and c) and add to 31 (the coefficient of x). These numbers are 16 and 15, allowing us to group and factor the expression into ( 5 x + 8 ) ( 2 x + 3 ) . Therefore, the final factored form is ( 5 x + 8 ) ( 2 x + 3 ) .
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