Problem 139 - Algebra and Trigonometry

Question

Factor completely. $$-x^{2}-4 x+5$$

Step-by-Step Solution

Answer

The complete factorization of the given quadratic expression is \( (x - 1)(x + 5) \)

Step 1: Determine the roots

Firstly, determine the roots of the equation by using the formula \( x = [-b ± sqrt(b^2 - 4ac)] / 2a .\) Here, \( a = -1, b = -4, c = 5 \). Thus applying the formula gives two values of \( x \), which are 1 and -5.

Step 2: Substitute the Roots

The roots obtained are substituted in place of \( m \) and \( n \) in the binomial expressions to get the performed factorization, which would look like \( (x - m)(x - n) = 0 \), after substituting the roots this becomes \( (x - 1)(x + 5) = 0 \)

Step 3: Factorize the Quadratic Expression

Now, multiplicate the factors \( (x - 1)(x + 5) \) to ensure the result is the original quadratic equation. This serves as a verification step. \( (x - 1)(x + 5) \) would yield \( -x^2 - 4x +5 \) on multiplication