Analyzing Quadratic Factorization and Graph Intersections

00:00 7 marks

Consider the quadratic function \(f(x) = ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants.

(a) For \(f(x) = 2x^2 – 8x + 6\), factorize the quadratic expression and find its roots analytically.

(b) Given the line \(g(x) = 4x – 2\), determine the intersection points between the graph of \(f(x)\) and the line \(g(x)\) using an analytical approach. Show all steps in your reasoning.

Part a

5 minutes 3 marks

Question

Factorize \(f(x) = 2x^2 – 8x + 6\) and determine its roots analytically.

Hint Hint

Hint

Consider factoring out the common factor first, then factorize the quadratic to find the roots.

Answer Answer
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Answer Answer

Complete Answer

Factorized form: \(2(x-1)(x-3)\); Roots: \(x = 1\) and \(x = 3\).

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Part b

7 minutes 4 marks

Question

Determine the intersection points of the function \(f(x) = 2x^2 – 8x + 6\) with the line \(g(x) = 4x – 2\) by solving \(f(x) = g(x)\).

Hint Hint

Hint

Set \(2x^2 – 8x + 6 = 4x – 2\) and solve for \(x\), then substitute the values back into \(g(x)\) to find \(y\).

Answer Answer
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Complete Answer

Intersection points: \((3+\sqrt{5},\,10+4\sqrt{5})\) and \((3-\sqrt{5},\,10-4\sqrt{5})\).

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