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The graph below represents function $k$ defined by $k(x) = x^2 + 3x$ - NSC Technical Mathematics - Question 2 - 2021 - Paper 1
Question 2
The graph below represents function $k$ defined by $k(x) = x^2 + 3x$. Describe the nature of the real roots of $k$. 2.2 Show that the roots of $x^2 + px - 2p^2 = 0... show full transcript
Worked Solution & Example Answer:The graph below represents function $k$ defined by $k(x) = x^2 + 3x$ - NSC Technical Mathematics - Question 2 - 2021 - Paper 1
Step 1
Describe the nature of the real roots of $k$.
Answer
To determine the nature of the real roots, we can analyze the graph of the function . Since the graph is a parabola that opens upwards and intersects the x-axis at two points (indicating two real roots), we conclude that the roots are real and rational. This is supported by the fact that the x-coordinates of the intersections are whole numbers, specifically and .
Step 2
Show that the roots of $x^2 + px - 2p^2 = 0$ are rational for $p$ real numbers.
Answer
To show that the roots are rational, we can calculate the discriminant of the quadratic equation. The discriminant is given by:
For our equation, where , , and , we substitute:
This simplifies to:
The discriminant is a perfect square, as . Since the discriminant is a perfect square, we conclude that the roots of the quadratic equation are rational for all real numbers .