AZ A curve has the equation $$ y=-2 x^{2}+4 x+3 $$. (i) Show that the lowest point on the curve has coordinates $$ (1,5) $$. (ii) Find the $$ x $$-coordinates of the points at which the curve intersects the $$ x $$-axis.

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**(i) Finding the vertex (which is the extremum point)**

The curve is given by
$$
y = -2x^2 + 4x + 3.
$$
We can rewrite the quadratic by completing the square.

1. Factor out the coefficient of $$ x^2 $$ from the first two terms:
   $$
   y = -2\left(x^2 - 2x\right) + 3.
   $$

2. Complete the square inside the parentheses. Recall that
   $$
   x^2 - 2x = \left(x - 1\right)^2 - 1.
   $$
   Thus,
   $$
   y = -2\left[\left(x-1\right)^2 - 1\right] + 3.
   $$

3. Distribute $$-2$$:
   $$
   y = -2\left(x-1\right)^2 + 2 + 3.
   $$

4. Combine constants:
   $$
   y = -2\left(x-1\right)^2 + 5.
   $$

The vertex form of a quadratic is
$$
y = a\left(x-h\right)^2 + k,
$$
which shows the vertex is at $$(h, k)$$. Here, the vertex is at $$(1, 5)$$.

---

**(ii) Finding the $$ x $$-coordinates where the curve intersects the $$ x $$-axis**

At the $$ x $$-axis, $$ y = 0 $$. Thus, set
$$
-2x^2 + 4x + 3 = 0.
$$

1. Multiply the equation by $$-1$$ to simplify:
   $$
   2x^2 - 4x - 3 = 0.
   $$

2. Use the quadratic formula where $$ a = 2 $$, $$ b = -4 $$, and $$ c = -3 $$:
   $$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
   $$

3. Substitute the values:
   $$
   x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4 \cdot 2 \cdot (-3)}}{2 \cdot 2} = \frac{4 \pm \sqrt{16 + 24}}{4}.
   $$

4. Simplify under the square root:
   $$
   \sqrt{16+24} = \sqrt{40} = 2\sqrt{10}.
   $$

5. Therefore,
   $$
   x = \frac{4 \pm 2\sqrt{10}}{4} = \frac{2 \pm \sqrt{10}}{2}.
   $$

The $$ x $$-coordinates of the intersections are
$$
x = \frac{2+\sqrt{10}}{2} \quad \text{and} \quad x = \frac{2-\sqrt{10}}{2}.
$$
**(i) The lowest point on the curve is at $$ (1, 5) $$.** **(ii) The curve intersects the $$ x $$-axis at $$ x = \frac{2+\sqrt{10}}{2} $$ and $$ x = \frac{2-\sqrt{10}}{2} $$.

AZ A curve has the equation
.
(i) Show that the lowest point on the
curve has coordinates .
(ii) Find the -coordinates of the
points at which the curve intersects
the -axis.

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AZ A curve has the equation . (i) Show that the lowest point on the curve has coordinates . (ii) Find the -coordinates of the points at which the curve intersects the -axis.