A-Level Edexcel Maths Pure Integration: The equation $x^2 + kx + (k + 3)

The equation $x^2 + kx + (k + 3) = 0$, where $k$ is a constant, has different real roots - Edexcel - A-Level Maths Pure - Question 9 - 2007 - Paper 1

Question 9

The equation $x^2 + kx + (k + 3) = 0$, where $k$ is a constant, has different real roots. (a) Show that $k^2 - 4k - 12 > 0$. (b) Find the set of possible values ... show full transcript

Worked Solution & Example Answer:The equation $x^2 + kx + (k + 3) = 0$, where $k$ is a constant, has different real roots - Edexcel - A-Level Maths Pure - Question 9 - 2007 - Paper 1

Step 1

(a) Show that $k^2 - 4k - 12 > 0$.

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Answer

To determine that the quadratic equation has different real roots, we must use the discriminant, defined as $b^2 - 4ac$ . For the given equation, we identify:

Coefficient of $x^2$ : $1$
Coefficient of $x$ : $k$
Constant term: $k + 3$

Now, substituting into the discriminant formula:

$D = k^2 - 4(1)(k + 3)$

Expanding this gives:

$D = k^2 - 4k - 12$

To ensure the roots are different and real, we need $D > 0$ , which translates to showing:

$k^2 - 4k - 12 > 0.$

This verifies that $k^2 - 4k - 12 > 0$ .

Step 2

(b) Find the set of possible values of $k$.

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Answer

To solve the inequality $k^2 - 4k - 12 > 0$ , we can factor it:

$(k - 6)(k + 2) > 0.$

Next, we find the critical points by setting each factor to zero:

$k - 6 = 0$ leads to $k = 6$
$k + 2 = 0$ leads to $k = -2$

Now, we will determine the intervals to test the inequality:

Interval: $(- ext{∞}, -2)$
Interval: $(-2, 6)$
Interval: $(6, ext{∞})$

Testing the intervals:

For $k = -3$ (in $(- ext{∞}, -2)$ ): $(-3 - 6)(-3 + 2) = (-9)(-1) > 0$ (True)
For $k = 0$ (in $(-2, 6)$ ): $(0 - 6)(0 + 2) = (-6)(2) < 0$ (False)
For $k = 7$ (in $(6, ext{∞})$ ): $(7 - 6)(7 + 2) = (1)(9) > 0$ (True)

Thus, $k^2 - 4k - 12 > 0$ is satisfied in the intervals:

$k ext{ in } (- ext{∞}, -2) ext{ or } (6, ext{∞}).$

Combining these gives us the set of possible values of $k$ :

$k ext{ such that } k < -2 ext{ or } k > 6.$

The equation $x^2 + kx + (k + 3) = 0$, where $k$ is a constant, has different real roots - Edexcel - A-Level Maths Pure - Question 9 - 2007 - Paper 1

Worked Solution & Example Answer:The equation $x^2 + kx + (k + 3) = 0$, where $k$ is a constant, has different real roots - Edexcel - A-Level Maths Pure - Question 9 - 2007 - Paper 1

(a) Show that $k^2 - 4k - 12 > 0$.

(b) Find the set of possible values of $k$.

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