(a) Expand and simplify.
\( 4(2x-1)-6(3-x) \)
(b) Factorise completely.
(i) \( 6x^2y+9xy \)
(ii) \( 4x^2-y^2+8x+4y \)
(c) Antonio travels 100 km at an average speed of x km/h.
He then travels a further 150 km at an average speed of (x + 10) km/h.
The time taken for the whole journey is 4 hours 20 minutes.
(i) Show that \( 13x^2 – 620x – 3000 = 0 \).
(ii) Solve \( 13x^2 – 620x – 3000 = 0 \) to find the speed Antonio travels for the first 100 km of the journey.
You must show all your working and give your answer correct to 1 decimal place.
▶️ Answer/Explanation
(a) \(14x – 22\) or \(2(7x – 11)\)
First expand both brackets: \(8x – 4 – 18 + 6x\). Then combine like terms to get \(14x – 22\).
(b)(i) \(3xy(2x + 3)\)
Factor out the greatest common factor \(3xy\) from both terms to get the simplified form.
(b)(ii) \((2x + y)(2x – y + 4)\)
Group terms: \((4x^2 – y^2) + (8x + 4y)\). Factor difference of squares first, then common factor in second group.
(c)(i) Proof shown
\[ \frac{100}{x} + \frac{150}{x+10} = 4\frac{1}{3} \]
Combine fractions: \[ \frac{100(x+10)+150x}{x(x+10)} = \frac{13}{3} \]
Simplify numerator: \[ \frac{250x+1000}{x(x+10)} = \frac{13}{3} \]
Cross multiply and expand: \[ 750x + 3000 = 13x^2 + 130x \]
Rearrange terms: \[ 13x^2 – 620x – 3000 = 0 \]
(c)(ii) 52.1 km/h
Use quadratic formula with \(a=13\), \(b=-620\), \(c=-3000\). Calculate discriminant and find positive root, rounding to one decimal place.
(a) Expand and simplify
(2p² – 3)(3p² – 2)
(b) s = ½(u + ν)t
(i) Find the value of s when u = 20, ν = 30 and t = 7
(ii) Rearrange the formula to write ν in terms of s, u and t
(c) Factorise completely.
(i) 2qt – 3t – 6 + 4q
(ii) x³ – 25x
▶️ Answer/Explanation
(a) 6p⁴ – 13p² + 6
Multiply each term: (2p²×3p²) + (2p²×-2) + (-3×3p²) + (-3×-2) = 6p⁴ – 4p² – 9p² + 6. Combine like terms.
(b)(i) 175
Substitute values: s = ½(20+30)×7 = ½×50×7 = 25×7 = 175.
(b)(ii) ν = (2s – ut)/t
Multiply both sides by 2: 2s = (u+ν)t. Expand: 2s = ut + νt. Isolate ν: ν = (2s – ut)/t.
(c)(i) (2q – 3)(t + 2)
Group terms: (2qt + 4q) + (-3t – 6). Factor each group: 2q(t+2) -3(t+2). Common factor (t+2).
(c)(ii) x(x + 5)(x – 5)
Factor out x: x(x² – 25). Recognize difference of squares: x² – 25 = (x+5)(x-5).