smc-750-40-Solve Quadratic

Algebra, MET1 2025 VCAA 9

Consider the functions

\(f: R \backslash\{1\} \rightarrow R, f(x)=\dfrac{w^2}{(x-1)^2}\)

and

\(g: R \rightarrow R, g(x)=(x-w)^2\)

where \(w \in R\).

If \(w=-3\), find the four solutions to \(f(x)=g(x)\). (3 marks)

--- 12 WORK AREA LINES (style=lined) ---
Consider the case where \(w>0\).
1. Find, in terms of \(w\), the coordinates of the minimum point of the graph of \(y=(x-1)(x-w)\). (2 marks)
  
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2. Hence, or otherwise, find the positive values of \(w\) for which \(f(x)=g(x)\) has exactly three solutions. (2 marks)
  
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a. \(\text{If} \ \ w=-3, \text{find 4 solutions to} \ \ f(x)=g(x):\)

\(\dfrac{(-3)^2}{(x-1)^2}=(x+3)^2\)

\((x+3)^2(x-1)^2=9\)

\((x+3)(x-1)= \pm 3\)

\(\text{Case 1}\)

\(x^2+2 x-3=3\)

\(x^2+2 x-6=0\)

\(x=-1 \pm \sqrt{7}\)

\(\text{Case 2}\)

\(x^2+2 x-3\)	\(=-3\)
\(x(x+2)\)	\(=0\)
\(x=0 \ \ \text{or} \ -2\)

b.i. \(\text{Min TP at} \ \left(\dfrac{w+1}{2}, -\dfrac{1}{4} (w-1)^2\right)\)

b.ii. \(\text{See worked solutions.}\)

Show Worked Solution

a. \(\text{If} \ \ w=-3, \text{find 4 solutions to} \ \ f(x)=g(x):\)

\(\dfrac{(-3)^2}{(x-1)^2}=(x+3)^2\)

\((x+3)^2(x-1)^2=9\)

\((x+3)(x-1)= \pm 3\)

\(\text{Case 1}\)

\(x^2+2 x-3=3\)

\(x^2+2 x-6=0\)

\(x=-1 \pm \sqrt{7}\)

♦♦ Mean mark (a) 34%.

\(\text{Case 2}\)

\(x^2+2 x-3\)	\(=-3\)
\(x(x+2)\)	\(=0\)
\(x=0 \ \ \text{or} \ -2\)

b.i. \(\text{Find minimum TP of} \ \ y=(x-1)(x-w)\)

\(\text{Axis of quadratic:}\ \ x=\dfrac{w+1}{2},\ \ (w>0)\)

\(y\)	\(=\left(\dfrac{w+1}{2}-1\right)\left(\dfrac{w+1}{2}-w\right)\)
	\(=\left(\dfrac{w-1}{2}\right)\left(\dfrac{1-w}{2}\right)\)
	\(=-\dfrac{1}{4}(w-1)^2\)

\(\therefore \ \text{Min TP at} \ \left(\dfrac{w+1}{2}, -\dfrac{1}{4} (w-1)^2\right)\)

♦♦ Mean mark (b.i) 39%.
♦♦♦ Mean mark (b.ii) 11%.

b.ii. \(\text{Solve for \(w\ (w>0)\) where \(f(x)=g(x)\):}\)

\(\dfrac{w^2}{(x-1)^2}\)	\(=(x-w)^2\)
\((x-w)^2(x-1)^2\)	\(=w^2\)
\([(x-w)(x-1)]^2-w^2\)	\(=0\)

\(\text{Difference between two squares:}\)

\([(x-w)(x-1)-w][(x-w)(x-1)+w]=0\)

\(\text{Case 1}\)

\((x-w)(x-1)-w\)	\(=0\)
\(x^2-x-w x+w-w\)	\(=0\)
\(x^2-(w+1) x\)	\(=0\)
\(x[x-(w+1)]\)	\(=0\)

\(x=0 \ \ \text{or} \ \ w+1\)

\(\text{Case 2}\)

\((x-w)(x-1)+w\)	\(=0\)
\(x^2-(w+1) x+2 w\)	\(=0\)

\(\text{1 solution if} \ \ \Delta=0:\)

\(\Delta\)	\(=[-(w+1)]^2-4 \times 1 \times 2 w\)
	\(=w^2+2 w+1-8 w\)
	\(=w^2-6 x+1\)

\(\text{If} \ \ \Delta=0:\)

\(w\)	\(=\dfrac{6 \pm \sqrt{(-6)^2-4 \times 1 \times 1}}{2}\)
	\(=\dfrac{6 \pm \sqrt{32}}{2}\)
	\(=3+2 \sqrt{2}\ \ (\text{reject}\ w=3-2 \sqrt{2}\ \ \text{as}\ \ w>0)\)

\(\therefore f(x)=g(x) \ \ \text{has exactly 3 solutions}\ (w>0).\)

Algebra, MET2 2007 VCAA 11 MC

The solution set of the equation `e^(4x) - 5e^(2x) + 4 = 0` over `R` is

A. `{1, 4}`

B. `{– 4, – 1})`

C. `{– 2, – 1, 1, 2})`

D. `{– log_e(2), 0, log_e(2)}`

E. `{0, log_e(2)}`

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`E`

Show Worked Solution

`text(Solve for)\ \ x\ \ text(on CAS:)`

`x = 0 \ or \ x = log_e(2)`

`=> E`

Algebra, MET2 2009 VCAA 5 MC

Let `f: R -> R,\ f (x) = x^2`

Which one of the following is not true?

`f(xy) = f (x) f (y)`
`f(x) - f(-x) = 0`
`f (2x) = 4 f (x)`
`f (x - y) = f(x) - f(y)`
`f (x + y) + f (x - y) = 2 (f (x) + f(y))`

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`D`

Show Worked Solution

`text(Solution 1)`

`text(Consider option)\ D:`

`f(x-y)`	`=(x-y)^2`
	`=x^2 -2xy+y^2`
`f(x)-f(y)`	`= x^2-y^2`
`:.f(x-y)`	`!=f(x)-f(y)`

`=>D`

`text(Solution 2)`

`text(Define)\ \ f(x) = x^2`

`text(Enter each functional equation on CAS)`

`text(until output does NOT read “true”.)`

`=> D`

Functions, MET1 2013 VCAA 9

The graph of `f(x) = (x-1)^2-2, x in [– 2, 2]`, is shown below. The graph intersects the `x`-axis where `x = a.`

Find the value of `a.` (1 mark)

Note: other parts of this question are out of the syllabus and not included.

--- 4 WORK AREA LINES (style=lined) ---

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`1-sqrt 2`

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`text(Find)\ \ x text(-intercept:)`

♦ Mean mark 50%.

`(x-1)^2-2`	`= 0`
`(x-1)^2`	`= 2`
`x-1`	`= +- sqrt 2`
`x`	`= 1 +- sqrt 2,\ \ a < 0`
`:. a`	`= 1-sqrt 2`

Graphs, MET2 2015 VCAA 7 MC

The range of the function `f:\ text{(−1, 2]} -> R,\ \ f(x) = -x^2 + 2x-3` is

`R`
`text{(−6, −3]}`
`text{(−6, −2]}`
`text{[−6, −3]}`
`text{[−6, −2]}`

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`C`

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`text(A sketch of the equation:)`

`y`	`=-x^2 + 2x-3`
`dy/dx`	`=-2x+2`

`=>\ text(Concave down with turning point when)\ \ x=1,`

`:. text(Range) = (-6,-2]`

`=> C`

Graphs, MET2 2012 VCAA 3 MC

The range of the function `f: text{[−2, 3)} -> R,\ f(x) = x^2 - 2x - 8` is

A. `R`

B. `text{(−9, −5]}`

C. `text{(−5, 0)}`

D. `text{[−9, 0]}`

E. `text{[−9, −5)}`

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`D`

Show Worked Solution

`text(Sketching the quadratic:)`

`:.\ text(Range) = [−9,0]`

`=> D`