Problem 138 - College Algebra Essentials

Question

$$\text { factor completely.}$$ $$-x^{2}-4 x+5$$

Step-by-Step Solution

Answer

The factored form of the equation \(-x^2 -4x +5\) is \((x + 5)(-x + 1)\).

Step 1: Determine the factors of the equation

In order to factor the quadratic equation, first identify the coefficients. For the given equation, \(a = -1\), \(b = -4\), and \(c = 5\). We need numbers that multiply to \(ac = -1 * 5 = -5\) and add to \(b = -4\). The numbers that fit this criteria are -5 and 1 because \(-5 * 1 = -5\) and \(-5 + 1 = -4\).

Step 2: Rewrite the middle term

The next step involves rewriting the middle term of the quadratic equation using the two factors discovered in Step 1. This gives us: \(-x^2 -5x + x + 5\).

Step 3: Group the terms and factor

Group the terms to factor by grouping: \((-x^2 - 5x) + (x + 5)\). Factor out an \(x\) from the first group to get \(-x(x + 5)\). Since the second group already has a common factor of 1, it can be written as is, i.e. \(+1(x + 5)\).

Step 4: Write the final factored form

After grouping and factoring, the equation becomes \(-x(x + 5) + 1(x + 5)\). Now, take out the common polynomial \(x + 5\) to see the final factored form which is \((x + 5)(-x + 1)\).