Step-by-step explanation:
To solve the quadratic equation \(2x^2 + 3x - 20 = 0\) by factorization, we need to find two numbers that multiply to -40 (the product of the coefficients of x^2 and the constant term) and add up to 3 (the coefficient of x).
The two numbers are 8 and -5 because 8 * (-5) = -40 and 8 + (-5) = 3.
Now, we rewrite the middle term using these two numbers:
\(2x^2 + 8x - 5x - 20 = 0\)
Factor by grouping:
\(2x(x + 4) - 5(x + 4) = 0\)
Factor out the common factor:
\((2x - 5)(x + 4) = 0\)
Set each factor to zero:
\(2x - 5 = 0\) or \(x + 4 = 0\)
Solve for x:
\(2x = 5 \Rightarrow x = \frac{5}{2}\)
\(x = -4\)
Therefore, the solutions to the equation \(2x^2 + 3x - 20 = 0\) are \(x = \frac{5}{2}\) and \(x = -4\).
