Question

mfm2p - culminating task
task 6: detour!!! day 3
for this challenge, your must complete two parts and solve problems using your algebra and proportional reasoning skills.
part 1:
a) find and circle 2 mistakes in the following solution: (2 marks)
$x^2 + 12x - 18 = 4x^2 + 12x - 9$
$x^2 - 4x^2 + 12x - 12x - 18 + 9 = 0$
$3x^2 - 9 = 0$
$x^2 = 9 - 3$
$x^2 = 6$
$x = 2.45$
b) solve for x. (3 marks) $x^2 - 4x + 60 = -2(-x^2 + 2x - 12)$

Explanation:

Part a)

Step1: Identify first error

When moving terms to left side, $-18$ becomes $+18$, so:
$x^2 - 4x^2 + 12x - 12x - 18 + 9 = 0$ is wrong. Correct: $x^2 - 4x^2 + 12x - 12x - 18 + 9 = -3x^2 -9 = 0$, not $3x^2 -9=0$ (sign error on $x^2$ term).

Step2: Identify second error

From $x^2 = 9 - 3$: $9-3=6$, but solving $-3x^2 -9=0$ gives $x^2 = -3$, so $x^2=6$ is invalid, and $x=2.45$ is wrong (no real solution here).

Part b)

Step1: Expand right-hand side

Distribute $-2$ to the parentheses:
$x^2 - 4x + 60 = 2x^2 -4x +24$

Step2: Move all terms to left side

Subtract $2x^2$, add $4x$, subtract $24$ from both sides:
$x^2 -4x +60 -2x^2 +4x -24 = 0$

Step3: Simplify the equation

Combine like terms:
$-x^2 + 36 = 0$

Step4: Rearrange to solve for $x^2$

$x^2 = 36$

Step5: Solve for $x$

Take square root of both sides:
$x = \pm 6$

Answer:

Part a)

1. Sign error: $3x^2 - 9 = 0$ should be $-3x^2 - 9 = 0$
2. Invalid solution step: $x^2 = 9 - 3$ is incorrect (the correct equation leads to $x^2 = -3$, which has no real solutions, so $x=2.45$ is wrong)

Part b)

$x = 6$ and $x = -6$