d) Step 1: Write the quadratic equation. $$ x^2 - 5x - 6 = 0 $$ Step 2: Factor the quadratic expression. Find two numbers that multiply to -6 and add to -5. These numbers are -6 and 1. $$ (x-6)(x+1) = 0 $$ Step 3: Set each factor equal to zero and solve for $x$. $$ x-6 = 0 \quad \text{or} \quad x+1 = 0 $$ $$ x = 6 \quad \text{or} \quad x = -1 $$ The solutions to the quadratic equation are $\boxed{x=6 \text{ or } x=-1}$. 4a) Step 1: Determine the real-life height of the room. The scale is 1:500, meaning 1 cm on the model represents 500 cm in real life. Model length = 5 cm. Real-life length = $5 \text{ cm} \times 500 = 2500 \text{ cm}$. Convert to meters: $2500 \text{ cm} = \frac{2500}{100} \text{ m} = 25 \text{ m}$. The height of the room is $\boxed{25 \text{ m}}$. Step 2: Determine the corresponding area on the model. The scale for length is $1:500$. The scale for area is $(1:500)^2 = 1:500^2 = 1:250000$. Real-life floor area = $36 \text{ m}^2$. Convert real-life area to cm$^2$: $1 \text{ m}^2 = (100 \text{ cm})^2 = 10000 \text{ cm}^2$. Real-life floor area = $36 \times 10000 \text{ cm}^2 = 360000 \text{ cm}^2$. Area on model = Real-life area $\times \frac{1}{250000}$. Area on model = $360000 \text{ cm}^2 \times \frac{1}{250000} = \frac{36}{25} \text{ cm}^2 = 1.44 \text{ cm}^2$. The corresponding area on the floor of the model is $\boxed{1.44 \text{ cm}^2}$. 4b) Step 1: Determine the corresponding volume on the model. The scale for length is $1:500$. The scale for volume is $(1:500)^3 = 1:500^3 = 1:125000000$. Real-life volume = $120 \text{ m}^3$. Convert real-life volume to cm$^3$: $1 \text{ m}^3 = (100 \text{ cm})^3 = 1000000 \text{ cm}^3$. Real-life volume = $120 \times 1000000 \text{ cm}^3 = 120000000 \text{ cm}^3$. Volume on model = Real-life volume $\times \frac{1}{125000000}$. Volume on model = $120000000 \text{ cm}^3 \times \frac{1}{125000000} = \frac{120}{125} \text{ cm}^3$. Step 2: Simplify the fraction. $$ \frac{120}{125} = \frac{24 \times 5}{25 \times 5} = \frac{24}{25} = 0.96 \text{ cm}^3 $$ The corresponding volume of the model is $\boxed{0.96 \text{ cm}^3}$. 5) Step 1: Identify similar triangles. Given that BA is parallel to CE, we can conclude that $\triangle CDE$ is similar to $\triangle BDA$. This is because: • $\angle DCE = \angle DBA$ (corresponding angles) • $\angle CED = \angle BAD$ (corresponding angles) • $\angle CDE = \angle BDA$ (common angle) Step 2: Set up the ratio of corresponding sides. From the similarity $\triangle CDE \sim \triangle BDA$, we have: $$ \frac{CE}{BA} = \frac{DE}{DA} $$ Step 3: Substitute the given values. We are given: BA = 9 cm CE = 4 cm AE = 3 cm Let DE = $x$. Then DA = DE + AE = $x + 3$. Substitute these into the ratio: $$ \frac{4}{9} = \frac{x}{x+3} $$ Step 4: Solve for $x$. Cross-multiply: $$ 4(x+3) = 9x $$ $$ 4x + 12 = 9x $$ Subtract $4x$ from both sides: $$ 12 = 9x - 4x $$ $$ 12 = 5x $$ Divide by 5: $$ x = \frac{12}{5} $$ $$ x = 2.4 \text{ cm} $$ The length of DE is $\boxed{2.4 \text{ cm}}$. That's 2 down. 3 left today — send the next one.