Quadratic equations form a cornerstone of upper-primary and secondary mathematics, appearing consistently in KAR TET Paper II. This topic tests both computational skills and conceptual understanding—you must be able to find roots using multiple methods, interpret the discriminant, and apply quadratic equations to word problems.
For TET preparation, focus on three areas: (1) solving equations using factorisation, completing the square, and the quadratic formula; (2) using the discriminant to determine the nature of roots without solving; and (3) understanding the relationship between roots and coefficients. Questions typically involve straightforward computation, but examiners also test whether candidates can identify which method suits a given problem and interpret results in real-world contexts.
Mastering quadratic equations also builds the foundation for coordinate geometry (parabolas), physics (projectile motion), and higher algebra—making this topic essential for both exam success and effective classroom teaching.
Key Concepts
**Standard form**: A quadratic equation must be written as ax² + bx + c = 0, where a ≠ 0. The condition a ≠ 0 is crucial—if a = 0, the equation becomes linear.
**Roots/solutions**: The values of x that satisfy the equation. A quadratic equation has exactly two roots (which may be equal, distinct, or complex).
**Discriminant (D or Δ)**: The expression b² − 4ac determines the nature of roots without actually solving the equation.
**Sum and product of roots**: If α and β are roots, then α + β = −b/a and αβ = c/a. This relationship helps verify answers and construct equations from given roots.
**Methods of solving**: Factorisation (quickest when applicable), completing the square (foundational method), and quadratic formula (universal method).
**Zero product property**: If the product of two factors equals zero, at least one factor must be zero. This underlies the factorisation method.
**Graphical interpretation**: Roots represent x-intercepts of the parabola y = ax² + bx + c. The discriminant tells us whether the parabola crosses, touches, or misses the x-axis.
Formulas / Key Facts
**Quadratic Formula** x = (−b ± √(b² − 4ac)) / 2a Use when factorisation is difficult or when exact roots are needed.
**Discriminant** D = b² − 4ac
D > 0 → Two distinct real roots
D = 0 → Two equal real roots (one repeated root)
D < 0 → No real roots (roots are complex)
**Sum of Roots** α + β = −b/a
**Product of Roots** αβ = c/a
**Forming Equation from Roots** If roots are α and β, the equation is: x² − (α + β)x + αβ = 0
**Completing the Square** To solve ax² + bx + c = 0: 1. Divide by a: x² + (b/a)x + c/a = 0 2. Move constant: x² + (b/a)x = −c/a 3. Add (b/2a)² to both sides 4. Factor left side as perfect square, solve
**Forgetting a ≠ 0** → Students write 0x² + 2x − 3 = 0 as a quadratic. Fix: Always verify the coefficient of x² is non-zero before applying quadratic methods.
**Sign errors in the quadratic formula** → Writing −b as just b, or mishandling the ± sign. Fix: Substitute values with their signs in parentheses: (−(3)) instead of −3.
**Incorrect discriminant calculation** → Computing b² − 4ac as b² − 4 × a × c without proper sign handling when c is negative. Fix: Use parentheses: 4(a)(c) and let the negative signs work through.
**Incomplete factorisation** → Finding x² − 5x + 6 = (x − 2)(x − 3) but reporting the roots as 2 and 3 without checking signs. Fix: Always substitute roots back into the original equation to verify.
**Confusing sum/product formulas** → Mixing up −b/a (sum) with c/a (product). Fix: Remember "Sum has the minus Sign" (−b/a for sum).
**Assuming no real roots means no solution** → Stating the equation "cannot be solved" when D < 0. Fix: Clarify that there are no real roots, but complex roots exist (though beyond upper-primary scope).
Quick Reference
Standard form: ax² + bx + c = 0 (a ≠ 0)
Quadratic formula: x = (−b ± √(b² − 4ac)) / 2a
D > 0: two distinct real roots; D = 0: equal roots; D < 0: no real roots
Sum of roots = −b/a; Product of roots = c/a
Always verify roots by substituting back into the original equation
Factorisation works best when roots are integers or simple fractions
You read the notes — now try one
If α and β are the roots of the equation x² - 5x + 6 = 0, then the value of α² + β² is:
Tap an option to check your answer.
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