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International Baccalaureate Mathematics

Functions

Syllabus Content

Solution of quadratic equations and inequalities. The quadratic formula. The discriminant ∆=b2-4ac and the nature of the roots, that is, two distinct real roots, two equal real roots, no real roots

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Furthermore

Official Guidance, clarification and syllabus links:

Using factorization, completing the square (vertex form), and the quadratic formula.

Solutions may be referred to as roots or zeros.

Example: For the equation \(3kx^2+2x+k=0\), find the possible values of \(k\), which will give two distinct real roots, two equal real roots or no real roots.

Formula Booklet:

Solutions of a quadratic equation

\(ax^2 + bx + c = 0 \implies x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} \)

Discriminant

\(\Delta = b^2 - 4ac \)

The solution of quadratic equations is a fundamental concept in algebra. A quadratic equation is typically in the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are coefficients. The solutions to these equations can be found using the quadratic formula:

$$ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}} $$

This formula calculates the roots of the quadratic equation. The nature of these roots is determined by the discriminant \( \Delta = b^2 - 4ac \). The discriminant reveals:

Example: Consider the quadratic equation \( 2x^2 - 4x - 6 = 0 \). To find the roots:

  1. Identify the coefficients: \( a = 2 \), \( b = -4 \), and \( c = -6 \).
  2. Compute the discriminant: \( \Delta = (-4)^2 - 4 \times 2 \times (-6) = 16 + 48 = 64 \).
  3. Since \( \Delta > 0 \), there are two distinct real roots.
  4. Apply the quadratic formula:
$$ x = \frac{{-(-4) \pm \sqrt{{64}}}}{{2 \times 2}} = \frac{{4 \pm 8}}{4} $$

Hence, the solutions are \( x = 3 \) and \( x = -1 \).

This video on Factorising Quadratic Functions and Equations is from Revision Village and is aimed at students taking the IB Maths AA Standard level course.

This video on Discriminant Test (Quadratics) is from Revision Village and is aimed at students taking the IB Maths AA Standard level course

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