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21 P1 Q2
2021 H2 Mathematics Paper 1 Question 2
Graphs and Transformations
Answers
(a)
(b)
x
=
1
+
5
{x=1 + \sqrt{5}}
x
=
1
+
5
or
x
=
−
1
+
7
{x = - 1 + \sqrt{7}}
x
=
−
1
+
7
Full solutions
(a)
(b)
x
2
−
5
=
2
x
−
1
x
2
−
5
=
−
(
2
x
−
1
)
\begin{align} && \quad x^2 - 5 &= 2x - 1 \\ && \quad x^2 - 5 &= -(2x - 1) \\ \end{align}
x
2
−
5
x
2
−
5
=
2
x
−
1
=
−
(
2
x
−
1
)
From
(
1
)
,
{(1),}
(
1
)
,
x
2
−
2
x
−
4
=
0
x^2 - 2 x - 4 = 0
x
2
−
2
x
−
4
=
0
x
=
2
±
2
2
−
4
(
−
4
)
(
1
)
2
=
2
±
20
2
=
2
±
2
5
2
=
1
±
5
\begin{align*} x &= \frac{2\pm\sqrt{2^2-4(-4)(1)}}{2} \\ &= \frac{2 \pm \sqrt{20}}{2} \\ &= \frac{2 \pm 2\sqrt{5}}{2} \\ & = 1 \pm \sqrt{5} \end{align*}
x
=
2
2
±
2
2
−
4
(
−
4
)
(
1
)
=
2
2
±
20
=
2
2
±
2
5
=
1
±
5
From
(
2
)
,
{(2),}
(
2
)
,
x
2
+
2
x
−
6
=
0
x^2 + 2 x - 6 = 0
x
2
+
2
x
−
6
=
0
x
=
−
2
±
2
2
−
4
(
−
6
)
(
1
)
2
=
−
2
±
28
2
=
−
2
±
2
7
2
=
−
1
±
7
\begin{align*} x &= \frac{-2\pm\sqrt{2^2-4(-6)(1)}}{2} \\ &= \frac{-2 \pm \sqrt{28}}{2} \\ &= \frac{-2 \pm 2\sqrt{7}}{2} \\ & = -1 \pm \sqrt{7} \end{align*}
x
=
2
−
2
±
2
2
−
4
(
−
6
)
(
1
)
=
2
−
2
±
28
=
2
−
2
±
2
7
=
−
1
±
7
From the sketch,
x
=
1
+
5
■
or
x
=
−
1
+
7
■
\begin{align*} && x & = 1 + \sqrt{5} \; \blacksquare \\ &\textrm{or}& x &= - 1 + \sqrt{7} \; \blacksquare \end{align*}
or
x
x
=
1
+
5
■
=
−
1
+
7
■
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