Score to pass: 70%
Related MCQS
1. End of Unit Test on Bivariate Data for S3 Students
2. End of Unit Test on Inverse and Composite Transformations in 2D for S3 Students
3. End of Unit Test on Enlargement and Similarity in 2D for S3 Students
4. End of Unit Test on Collinear Points and Orthogonal Vectors for S3 Students
5. End of Unit Test on Right Angled Triangles for S3 Students
6. End of Unit Test on Percentage Interest and Proportion for S3 Students
7. End of Unit Test on Linear and Quadratic Functions for S3 Students
8. End of Unit Test on Simultaneous Equations and Inequalities for S3 Students
9. End of Unit Test on Algebraic Fractions for S3 Students
10. End of Unit Test on Number Bases for S3 Students
11. End of Unit Test on Problems on Sets for S3 Students
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Quadratic Equations Unit Summary
1. A quadratic equation: This is an equation that is of the form \(ax^{2} + bx + c = 0\) where \(a\), \(b\) and \(c\) are constants and \(a \neq 0\). Examples of quadratic equations are \(x^2 + 9x + 14 = 0\), \(u^2 – 5u + 4 = 0\), \(7 – 6r + r^2 = 0\) etc.
2. Quadratic equations can be solved by the following methods.
(1) Factorisation method
(2) Graphical method
(3) Completing squares method
(4) Quadratic formula method
(5) Synthetic division method
3. In order to solve a quadratic equation, the quadratic expression is factorized so that the equation is in the form \((x + a)(x + b) = 0\). Then either \((x + a) = 0\) or \((x + b) = 0\) Thus \(x = –a\) and \(x = –b\).
4. In a quadratic function graph, the x-coordinate of the point where the graph cuts x-axis gives the solution to the quadratic equation represented by the function.
(a) When the graph cuts the x-axis at one point, then the equation has one repeated solution.
(b) When the graph cuts x-axis at two points, then the equation has two different solutions.
(c) When the graph does not cut x-axis at any point, then the equation has no solution in the f ield of real numbers.
5. Before solving any quadratic equation by completing squares method, it is better to understand what perfect squares are. For example \(x^2 + 6x + 9 = (x + 3)^2\) is a perfect square. Remember that \((a + b)^2 = a^2 + 2ab + b^2\) and that \((a – b)^2 = a^2 – 2ab + b^2\).
6. The quadratic formula \(x =\frac{ –b ± \sqrt{b^2 – 4ac}}{2a}\) is also suitable for solving quadratic equation \(ax^2 + bx + c = 0\) provide the coefficients \(a\), \(b\) and \(c\) are all known.
7. The quadratic equation \(ax^2 + bx + c = 0\) can also be solved by synthetic division method as long as the value x = f is the factor of constant c. We follow the trend below.
For vertical patterns, add terms while for the diagonal patterns, multiply by factor f. Hence, on dividing the expression \(ax^2 + bx + c\) by \((x – f)\), we should get the other factor (quotient) as \(ax + (b + af)\). Thus the quadratic equation \(ax^2 + bx + c = (x – f)(ax + (b + af))\) in factorised form. We therefore solve the values of \(x\) from the factors \((x – f)(ax + (b + af)) = 0\). Meaning \(x – f = 0\) and \(ax + (b + af) = 0\).
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