Solve \(2 \log _3(x-4)+\log _3(x)=2\) for \(x\). (4 marks) --- 8 WORK AREA LINES (style=lined) ---
\(1^3-8(1)^2+16(1)-9=0\) \(\therefore\ x-1\ \text{is a factor} \) \((x-1)(x^2-7x+9)=0\) \(\text{Using quadratic formula to solve}\ \ x^2-7x+9=0:\) \( x=1, \dfrac{7- \sqrt{13}}{2}, \dfrac{7 + \sqrt{13}}{2}\) \(\therefore\ \dfrac{7 + \sqrt{13}}{2}\ \text{ is the only possible solution.}\)
\(2\log_3(x-4)+\log_3(x)\)
\(=2\)
\(\log_3x(x-4)^2\)
\(=2\)
\(x(x-4)^2\)
\(=3^2\)
\(x(x^2-8x+16)-9\)
\(=0\)
\(x^3-8x^2+16x-9\)
\(=0\)
\(\text{Find a factor}\ \ \Rightarrow\ \ \text{Test}\ \ x=1:\)
\(x\)
\(=\dfrac{-(-7)\pm\sqrt{(-7)^2-4(1)(9)}}{2(1)}\)
\(=\dfrac{7\pm \sqrt{49-36}}{2}\)
\(=\dfrac{7\pm \sqrt{13}}{2}\)
\(\text{For }\log_3(x-4)\ \text{to exist}\ x>4\)
Algebra, MET2 2011 VCAA 18 MC
The equation `x^3 - 9x^2 + 15x + w = 0` has only one solution for `x` when
A. `−7 < w < 25`
B. `w <= −7`
C. `w >=25`
D. `w < −7` or `w > 25`
E. `w > 1`
Show Worked Solution
| `y` | `=x^3 – 9x^2 + 15x + w` |
| `y′` | `=3x^2 – 18x + 15` |
| `=3(x-5)(x-1)` |
`text(Stationary points at)\ \ x=5 and 1.`
`text{Sketch the graph}\ \ (w=0),`
`text(By inspection, one solution occurs when)`
`w > 25\ \ text{(shift curve up > 25), or}`
`w < −7\ \ text{(shift curve down > 7)`
`=> D`
Graphs, MET2 2015 VCAA 17 MC
A graph with rule `f(x) = x^3 - 3x^2 + c`, where `c` is a real number, has three distinct `x`-intercepts
The set of all possible values of `c` is
A. `R`
B. `R^+`
C. `{0, 4}`
D. `(0, 4)`
E. `text{(−∞, 4)}`
Show Worked Solution
`text(Consider the graph of)\ \ f(x) = x^3 – 3x^2`
`text(For three)\ xtext(-intercepts, translate up)`
`text{between (0, 4) units}`
`c ∈ (0,4)`
`=> D`
Algebra, MET2 2015 VCAA 6 MC
For the polynomial `P(x) = x^3 - ax^2 - 4x + 4,\ \ P(3) = 10,` the value of `a` is
A. `− 3`
B. `− 1`
C. `1`
D. `3`
E. `10`
Calculus, MET2 2013 VCAA 21 MC
The cubic function `f: R -> R, f(x) = ax^3-bx^2 + cx`, where `a, b` and `c` are positive constants, has no stationary points when
- `c > b^2/(4a)`
- `c < b^2/(4a)`
- `c < 4b^2a`
- `c > b^2/(3a)`
- `c < b^2/(3a)`