Equations & Inequalities
Quadratic, simultaneous and exponential equations
Quadratic, Simultaneous and Exponential Equations
In Grade 11, we solve:
• Quadratic equations: ax² + bx + c = 0 (factorise, complete the square, or use the quadratic formula)
• Simultaneous equations: one linear and one quadratic
• Exponential equations: equations with variables in the exponent
Example
Quadratic Equation
Solve x² − 5x + 6 = 0:
Factorise: (x − 2)(x − 3) = 0
x = 2 or x = 3
Solve 2x² + 3x − 5 = 0 using the formula:
x = [−3 ± √(9 + 40)] / 4 = (−3 ± 7)/4
x = 1 or x = −5/2
Example
Simultaneous Equations
y = x + 1 ... (1)
x² + y² = 13 ... (2)
Substitute (1) into (2):
x² + (x+1)² = 13
x² + x² + 2x + 1 = 13
2x² + 2x − 12 = 0
x² + x − 6 = 0
(x+3)(x−2) = 0
x = −3, y = −2 or x = 2, y = 3
Note
Remember
Quadratic formula: x = [−b ± √(b²−4ac)] / 2a. If b²−4ac < 0, there are no real roots. For exponential equations, express both sides with the same base, then equate exponents.
Key Vocabulary
Quadratic equationAn equation of the form ax² + bx + c = 0
Discriminantb² − 4ac; determines the nature of roots
Simultaneous equationsTwo or more equations solved together
RootA solution to an equation
Quadratic formulax = [−b ± √(b²−4ac)] / 2a
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Quadratic equation
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