NSC Technical Mathematics Equations and inequalities: Given: $q = \frac{3 \pm \sqrt{

Given: $q = \frac{3 \pm \sqrt{1 - 3k}}{k - 4}$ Determine for which value(s) of $k$ will $q$: 2.1.1 Have equal roots 2.1.2 Be undefined Given the equation: $4x^2 + 3x + p = 0$ 2.2 Complete the following statement: 2.2.1 If the roots are non-real, then the value of the discriminant is .. - NSC Technical Mathematics - Question 2 - 2023 - Paper 1

Question 2

$Given:---$q-=-\frac{3-\pm-\sqrt{1---3k}}{k---4}$---Determine-for-which-value(s)-of-$k$-will-$q$:---2.1.1-Have-equal-roots---2.1.2-Be-undefined----Given-the-equation:---$4x^2-+-3x-+-p-=-0$---2.2-Complete-the-following-statement:---2.2.1-If-the-roots-are-non-real,-then-the-value-of-the-discriminant-is-..-NSC Technical Mathematics-Question 2-2023-Paper 1.png$

Given: $q = \frac{3 \pm \sqrt{1 - 3k}}{k - 4}$ Determine for which value(s) of $k$ will $q$: 2.1.1 Have equal roots 2.1.2 Be undefined Given the equation:... show full transcript

Worked Solution & Example Answer:Given: $q = \frac{3 \pm \sqrt{1 - 3k}}{k - 4}$ Determine for which value(s) of $k$ will $q$: 2.1.1 Have equal roots 2.1.2 Be undefined Given the equation: $4x^2 + 3x + p = 0$ 2.2 Complete the following statement: 2.2.1 If the roots are non-real, then the value of the discriminant is .. - NSC Technical Mathematics - Question 2 - 2023 - Paper 1

Step 1

2.1.1 Have equal roots

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Answer

To determine when the quadratic has equal roots, we need the discriminant to be zero. For the expression under the square root, we set:

$1 - 3k = 0$

Solving for $k$ , we get: $3k = 1 \implies k = \frac{1}{3}$ Thus, $k$ must be $\frac{1}{3}$ for $q$ to have equal roots.

Step 2

2.1.2 Be undefined

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Answer

The value of $q$ will be undefined when the denominator of the fraction equals zero. Hence:

$k - 4 = 0 \implies k = 4$ So, $k = 4$ will make $q$ undefined.

Step 3

2.2.1 If the roots are non-real, then the value of the discriminant is ...

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Answer

If the roots are non-real, then the discriminant must be less than zero:

$\Delta < 0$

Step 4

2.2.2 Determine the value of p, for which the roots of the equation will be non-real.

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Answer

The discriminant for the quadratic equation $4x^2 + 3x + p = 0$ is given by:

$\Delta = b^2 - 4ac$

Substituting the values:

$\Delta = (3)^2 - 4(4)(p)$

This simplifies to:

$9 - 16p < 0$

Solving for $p$ :

$9 < 16p \implies p > \frac{9}{16}$ Thus, the roots will be non-real if $p > \frac{9}{16}$ .

2.1.1 Have equal roots

2.1.2 Be undefined

2.2.1 If the roots are non-real, then the value of the discriminant is ...

2.2.2 Determine the value of p, for which the roots of the equation will be non-real.

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