Answer :
Let's solve the problem step-by-step, addressing each part in detail:
### Part (a)
Given the equivalence [tex]\(x^2 + 2x + 3 \equiv (x + a)^2 + b\)[/tex]:
1. Expand the right-hand side:
[tex]\[ (x + a)^2 + b = x^2 + 2ax + a^2 + b \][/tex]
2. Equate the coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant terms from both sides:
[tex]\[ x^2 + 2x + 3 = x^2 + 2ax + a^2 + b \][/tex]
By comparing coefficients:
- For [tex]\(x^2\)[/tex]: The coefficient is [tex]\(1\)[/tex]; hence, [tex]\(1 = 1\)[/tex] (This is just a sanity check)
- For [tex]\(x\)[/tex]: The coefficient is [tex]\(2\)[/tex]; hence, [tex]\(2a = 2\)[/tex]
[tex]\[ a = 1 \][/tex]
- For the constant term: [tex]\(3 = a^2 + b\)[/tex]
[tex]\[ 3 = 1^2 + b \implies 3 = 1 + b \implies b = 2 \][/tex]
Thus, the values are:
[tex]\[ a = 1, \quad b = 2 \][/tex]
### Part (b)
To sketch the graph of [tex]\(y = x^2 + 2x + 3\)[/tex], we use its vertex form:
[tex]\[ y = (x + 1)^2 + 2 \][/tex]
This shows us the graph is a parabola with:
- Vertex at [tex]\((-1, 2)\)[/tex]
- Y-intercept: Set [tex]\(x = 0\)[/tex]:
[tex]\[ y = 0^2 + 2 \cdot 0 + 3 = 3 \][/tex]
### Parabola Sketch
1. Vertex: [tex]\((-1, 2)\)[/tex]
2. Y-intercept: [tex]\(y = 3\)[/tex]
3. X-intercepts: Since we shall find no real roots in the next part, the graph does not intersect the x-axis.
### Part (c)
To find the value of the discriminant for [tex]\(x^2 + 2x + 3\)[/tex]:
The discriminant [tex]\(\Delta\)[/tex] for a quadratic equation [tex]\(ax^2 + bx + c\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For [tex]\(x^2 + 2x + 3\)[/tex]:
[tex]\[ a = 1, \quad b = 2, \quad c = 3 \][/tex]
So, the discriminant is:
[tex]\[ \Delta = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8 \][/tex]
The discriminant [tex]\(\Delta = -8 < 0\)[/tex] indicates that the quadratic equation has no real roots. This confirms that the parabola does not intersect the x-axis, as anticipated in part (b).
### Part (d)
To find the set of possible values for [tex]\(k\)[/tex] in [tex]\(x^2 + kx + 3 = 0\)[/tex] such that there are no real roots, we require the discriminant to be less than zero:
[tex]\[ \Delta = k^2 - 4ac < 0 \][/tex]
For [tex]\(x^2 + kx + 3\)[/tex]:
[tex]\[ a = 1, \quad b = k, \quad c = 3 \][/tex]
The discriminant is:
[tex]\[ \Delta = k^2 - 12 < 0 \][/tex]
We solve this:
[tex]\[ k^2 < 12 \implies -\sqrt{12} < k < \sqrt{12} \][/tex]
Hence, the possible values of [tex]\(k\)[/tex] in surd form are:
[tex]\[ k \in (-2\sqrt{3}, 2\sqrt{3}) \][/tex]
### Full Summary
1. The constants are [tex]\(a = 1\)[/tex] and [tex]\(b = 2\)[/tex].
2. The graph of [tex]\(y = x^2 + 2x + 3\)[/tex] is a parabola with vertex [tex]\((-1, 2)\)[/tex] and y-intercept [tex]\(3\)[/tex]; it does not intersect the x-axis.
3. The discriminant of [tex]\(x^2 + 2x + 3\)[/tex] is [tex]\(\Delta = -8\)[/tex], confirming the lack of real roots.
4. The possible values of [tex]\(k\)[/tex] for which the equation [tex]\(x^2 + kx + 3 = 0\)[/tex] has no real roots are [tex]\(k \in (-2\sqrt{3}, 2\sqrt{3})\)[/tex].
### Part (a)
Given the equivalence [tex]\(x^2 + 2x + 3 \equiv (x + a)^2 + b\)[/tex]:
1. Expand the right-hand side:
[tex]\[ (x + a)^2 + b = x^2 + 2ax + a^2 + b \][/tex]
2. Equate the coefficients of [tex]\(x^2\)[/tex], [tex]\(x\)[/tex], and the constant terms from both sides:
[tex]\[ x^2 + 2x + 3 = x^2 + 2ax + a^2 + b \][/tex]
By comparing coefficients:
- For [tex]\(x^2\)[/tex]: The coefficient is [tex]\(1\)[/tex]; hence, [tex]\(1 = 1\)[/tex] (This is just a sanity check)
- For [tex]\(x\)[/tex]: The coefficient is [tex]\(2\)[/tex]; hence, [tex]\(2a = 2\)[/tex]
[tex]\[ a = 1 \][/tex]
- For the constant term: [tex]\(3 = a^2 + b\)[/tex]
[tex]\[ 3 = 1^2 + b \implies 3 = 1 + b \implies b = 2 \][/tex]
Thus, the values are:
[tex]\[ a = 1, \quad b = 2 \][/tex]
### Part (b)
To sketch the graph of [tex]\(y = x^2 + 2x + 3\)[/tex], we use its vertex form:
[tex]\[ y = (x + 1)^2 + 2 \][/tex]
This shows us the graph is a parabola with:
- Vertex at [tex]\((-1, 2)\)[/tex]
- Y-intercept: Set [tex]\(x = 0\)[/tex]:
[tex]\[ y = 0^2 + 2 \cdot 0 + 3 = 3 \][/tex]
### Parabola Sketch
1. Vertex: [tex]\((-1, 2)\)[/tex]
2. Y-intercept: [tex]\(y = 3\)[/tex]
3. X-intercepts: Since we shall find no real roots in the next part, the graph does not intersect the x-axis.
### Part (c)
To find the value of the discriminant for [tex]\(x^2 + 2x + 3\)[/tex]:
The discriminant [tex]\(\Delta\)[/tex] for a quadratic equation [tex]\(ax^2 + bx + c\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
For [tex]\(x^2 + 2x + 3\)[/tex]:
[tex]\[ a = 1, \quad b = 2, \quad c = 3 \][/tex]
So, the discriminant is:
[tex]\[ \Delta = 2^2 - 4 \cdot 1 \cdot 3 = 4 - 12 = -8 \][/tex]
The discriminant [tex]\(\Delta = -8 < 0\)[/tex] indicates that the quadratic equation has no real roots. This confirms that the parabola does not intersect the x-axis, as anticipated in part (b).
### Part (d)
To find the set of possible values for [tex]\(k\)[/tex] in [tex]\(x^2 + kx + 3 = 0\)[/tex] such that there are no real roots, we require the discriminant to be less than zero:
[tex]\[ \Delta = k^2 - 4ac < 0 \][/tex]
For [tex]\(x^2 + kx + 3\)[/tex]:
[tex]\[ a = 1, \quad b = k, \quad c = 3 \][/tex]
The discriminant is:
[tex]\[ \Delta = k^2 - 12 < 0 \][/tex]
We solve this:
[tex]\[ k^2 < 12 \implies -\sqrt{12} < k < \sqrt{12} \][/tex]
Hence, the possible values of [tex]\(k\)[/tex] in surd form are:
[tex]\[ k \in (-2\sqrt{3}, 2\sqrt{3}) \][/tex]
### Full Summary
1. The constants are [tex]\(a = 1\)[/tex] and [tex]\(b = 2\)[/tex].
2. The graph of [tex]\(y = x^2 + 2x + 3\)[/tex] is a parabola with vertex [tex]\((-1, 2)\)[/tex] and y-intercept [tex]\(3\)[/tex]; it does not intersect the x-axis.
3. The discriminant of [tex]\(x^2 + 2x + 3\)[/tex] is [tex]\(\Delta = -8\)[/tex], confirming the lack of real roots.
4. The possible values of [tex]\(k\)[/tex] for which the equation [tex]\(x^2 + kx + 3 = 0\)[/tex] has no real roots are [tex]\(k \in (-2\sqrt{3}, 2\sqrt{3})\)[/tex].
