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Are these solutions correct if not plz fix them

*Problem Statement*

In the diagram, the graphs of $f(x) = -x^2 + 5x + 6$ and $g(x)$ intersect the graph of $f(x)$ at $B$ and $C$. $g(x)$ is perpendicular to the x-axis with points $P$ and $Q$ on $f(x)$ and $g(x)$ respectively. $M$ is the TP of $f(x)$.

*Step 1: Calculate the Coordinates of B and C*

To find the coordinates of $B$ and $C$, we need to determine where $f(x)$ intersects the x-axis.

- Set $f(x) = 0$: $-x^2 + 5x + 6 = 0$
- Solve for $x$: $x^2 - 5x - 6 = 0$
- Factorize: $(x - 6)(x + 1) = 0$
- Roots: $x = 6$ or $x = -1$

Therefore, the coordinates of $B$ and $C$ are $(-1, 0)$ and $(6, 0)$ respectively.

*Step 2: If $PQ = 5$ units, Calculate the Length of OR*

Since $g(x)$ is perpendicular to the x-axis, it is a vertical line. The distance between $P$ and $Q$ is the difference in their y-coordinates.

- $PQ = f(x) - g(x) = 5$
- At point $P$, $x = x_P$, so $f(x_P) - g(x_P) = 5$
- Since $g(x_P) = g(x_Q)$, we have $f(x_P) - g(x_Q) = 5$

Given that $g(x) = -x^2 + 4x$ (from the table), we can substitute $x = 2$ into both functions to check if $PQ = 5$:

- $f(2) = -(2)^2 + 5(2) + 6 = -4 + 10 + 6 = 12$
- $g(2) = -(2)^2 + 4(2) = -4 + 8 = 4$
- $PQ = f(2) - g(2) = 12 - 4 = 8 \neq 5$

Let's try $x = 3$:

- $f(3) = -(3)^2 + 5(3) + 6 = -9 + 15 + 6 = 12$
- $g(3) = -(3)^2 + 4(3) = -9 + 12 = 3$
- $PQ = f(3) - g(3) = 12 - 3 = 9 \neq 5$

Let's try $x = 4$:

- $f(4) = -(4)^2 + 5(4) + 6 = -16 + 20 + 6 = 10$
- $g(4) = -(4)^2 + 4(4) = -16 + 16 = 0$
- $PQ = f(4) - g(4) = 10 - 0 = 10 \neq 5$

Let's try $x = 1$:

- $f(1) = -(1)^2 + 5(1) + 6 = -1 + 5 + 6 = 10$
- $g(1) = -(1)^2 + 4(1) = -1 + 4 = 3$
- $PQ = f(1) - g(1) = 10 - 3 = 7 \neq 5$

Let's try $x = 5$:

- $f(5) = -(5)^2 + 5(5) + 6 = -25 + 25 + 6 = 6$
- $g(5) = -(5)^2 + 4(5) = -25 + 20 = -5$
- $PQ = f(5) - g(5) = 6 - (-5) = 11 \neq 5$

Let's try $x = 0$:

- $f(0) = -(0)^2 + 5(0) + 6 = 6$
- $g(0) = -(0)^2 + 4(0) = 0$
- $PQ = f(0) - g(0) = 6 - 0 = 6 \neq 5$

Let's try $x = 2.5$:

- $f(2.5) = -(2.5)^2 + 5(2.5)

1 calculate:
Co ordinates of M
Maximum length of PQ between bend s

1 answer

Your solutions appear to have several errors and misunderstandings. Let’s go through your steps and correct them:

Problem Statement

You have two functions \( f(x) = -x^2 + 5x + 6 \) and \( g(x) \). You are looking for intersections of \( f(x) \) with the x-axis, the coordinates of points at which \( g(x) \) intersects \( f(x) \) vertically, and other related calculations.

Step 1: Calculate the Coordinates of \( B \) and \( C \)

You intend to find the points where \( f(x) \) intersects the x-axis. This requires \( f(x) = 0 \).

  1. Starting with the right equation: \[ -x^2 + 5x + 6 = 0 \] This is already correct.

  2. Rearranging: \[ x^2 - 5x - 6 = 0 \]

  3. Factoring: \[ (x - 6)(x + 1) = 0 \] So, the roots are: \[ x = 6 \quad \text{and} \quad x = -1 \]

  4. Thus, the coordinates of \( B \) and \( C \) as intersections with the x-axis are: \[ B(-1, 0) \quad \text{and} \quad C(6, 0) \] Your calculations for the coordinates of \( B \) and \( C \) are correct.

Step 2: Calculate the Length of \( PQ \)

You need to find the points \( P \) and \( Q \) on \( f(x) \) and \( g(x) \) respectively, with \( PQ = 5 \).

  1. The expression \( PQ = f(x) - g(x) = 5 \) is somewhat misleading as it lacks clarity on how you are defining \( g(x) \). You mention that \( g(x) \) is perpendicular to the x-axis and therefore represents a vertical line \( x = k \) where \( k \) will be the x-coordinate you need to substitute.

  2. You provided another function \( g(x) = -x^2 + 4x \) to evaluate points. This needs to be stated clearly at the beginning.

  3. You correctly checked various points for \( P \) based on \( f(x) \) and computed \( PQ \). However, you seem to be looking for a specific \( x \) value that satisfies \( PQ = 5 \).

Corrected Solving Method

To find coordinates \( P \) and \( Q \):

  1. From \( g(x) = -x^2 + 4x \): Let’s evaluate the distance \( f(x) - g(x) \) and set it equal to \( 5 \):

    \[ f(x) - g(x) = (-x^2 + 5x + 6) - (-x^2 + 4x) = (5x - 4x) + 6 = x + 6 \]

    Set \( x + 6 = 5 \): \[ x = -1 \]

  2. Now substitute \( x = -1 \) back into \( f(x) \) and \( g(x) \): \[ f(-1) = -(-1)^2 + 5(-1) + 6 = -1 - 5 + 6 = 0 \] \[ g(-1) = -(-1)^2 + 4(-1) = -1 -4 = -5 \]

Thus, the coordinates become:

  • \( P(-1, 0) \)
  • \( Q(-1, -5) \)

Maximum Length of \( PQ \)

To find the maximum distance, take into consideration that tiny changes in \( x \) lead to new distances \( PQ \), especially as we check vertices of limitations of \( f(x) \) and \( g(x) \). Continuously evaluate both across their domain until these curves cannot exceed 5 apart.

Final Considerations

Check physical relevance and limitations for more points or different vertical approaches to height outstretching the function pairs directly.

The overall work demonstrates the necessity of clear notation and verified functional evaluations to correctly deduce intersections or distance measures required by the problem statement.