$$ \left. \begin{array} { l } { g ^ { 2 } - 5 g - 6 } \ { = ( g - 3 ) ( g - 2 ) } \end{array} ight. $$

Question

$$ \left. \begin{array} { l } { g ^ { 2 } - 5 g - 6 } \ { = ( g - 3 ) ( g - 2 ) } \end{array} ight. $$

UpStudy AI · Accepted Answer

We start with the expression

$$
g^2 - 5g - 6 = (g-3)(g-2).
$$

**Step 1. Expand the Right-Hand Side**

Expand $$(g-3)(g-2)$$ by multiplying the factors:

$$
(g-3)(g-2) = g^2 - 2g - 3g + 6 = g^2 - 5g + 6.
$$

**Step 2. Compare Both Sides**

The left-hand side is

$$
g^2 - 5g - 6,
$$

and the right-hand side (after expansion) is

$$
g^2 - 5g + 6.
$$

Since the quadratic and linear terms match, compare the constant terms:

- Left-hand side constant: $$-6$$
- Right-hand side constant: $$+6$$

These constants are not equal.

**Step 3. Conclude the Evaluation**

Setting the two sides equal gives:

$$
g^2 - 5g - 6 = g^2 - 5g + 6.
$$

Subtract $$g^2 - 5g$$ from both sides:

$$
-6 = 6,
$$

which is a contradiction. This shows that the two expressions are not equal.

**Step 4. Correct Factorization of $$g^2 - 5g - 6$$**

To factor $$g^2 - 5g - 6$$, we need two numbers that multiply to $$-6$$ (the constant term) and add to $$-5$$ (the coefficient of $$g$$). The pair $$-6$$ and $$1$$ meets these conditions:

$$
-6 \times 1 = -6 \quad \text{and} \quad -6 + 1 = -5.
$$

Thus, the correct factorization is:

$$
g^2 - 5g - 6 = (g-6)(g+1).
$$

**Conclusion**

The given equation

$$
g^2-5g-6 = (g-3)(g-2)
$$

does not hold because

$$
(g-3)(g-2)=g^2-5g+6,
$$

which differs from $$g^2-5g-6$$. The correct factorization of $$g^2-5g-6$$ is

$$
(g-6)(g+1).
$$
The equation $$ g^2 - 5g - 6 = (g-3)(g-2) $$ is incorrect because expanding the right side gives $$ g^2 - 5g + 6 $$, which does not match the left side. The correct factorization of $$ g^2 - 5g - 6 $$ is $$ (g-6)(g+1) $$.

\( \left. \begin{array} { l } { g ^ { 2 } - 5 g - 6 } \\ { = ( g - 3 ) ( g - 2 ) } \end{array} \right. \)

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