J17

5. (x + 13) 2 = 0 6. x 2 + x – 12 = 0
7. x 2 + 9x + 14 = 0 8. x 2 – 11x – 12 = 0
9. u 2 + 6u + 9 = 0 10. t 2 – t – 42 = 0
11. a 2 – 2a + 1 = 0 12. y 2 + 8y = 0
13. x 2 + 10x = 24 14. x 2 = 4x – 3
15. v 2 – 36 = 0 16. 4x 2 – 9 = 0
17. 1 – y 2 = 0 18. 81 – 16q 2 = 0
19. x 2 = 6x 20. 5x 2 = 45
21. 6x(2x + 3) = 0 22. -4x(3x – 5) = 0
23. (3x – 2)(2x + 1) = 0 24. 6x 2 – 29x + 35 = 0
25. 6x 2 – x + 1 = 0 26. (4x – 3)(3x – 4) = 0
27. 4y 2 – 48 = 16 28. 3y 2 – 24 = 3
Quadratic equations
Introduction
When an equation of the form y = ax + b (a ≠ 0 and a, b are constants) is plotted on a
Cartesian plane, the graph is a straight line. Hence we call an expression of the form ax
+ b a linear expression and an equation of the form ax + b = 0 is called a linear equation.
Thus 3x + 5 is a linear expression,
y = 3x + 5 represents a straight line graph.
When an equation of the form y = ax 2 + bx + c (a ≠ 0; a, b and c are constants) is plotted
on a Cartesian plane, the graph is a curve known as a parabola.
An expression of the form ax 2 + bx + c is called a quadratic expression.
An equation of the form ax 2 + bx + c = 0 is called a quadratic equation.
Thus, 2x 2 + 3x + 1, 3x 2 + 5 and x 2 + 3x are quadratic expressions, whereas,
2x 2 + 3x + 1 = 0; 3x 2 + 5 = 0 and x 2 + 3x = 0 are quadratic equations.
Solving quadratic equations by factors method
We solve an equation by finding the value of the unknown. For any two numbers, p and
q, if pq = 0, then, either:
(d) p = 0 which means 0 × q = 0
(e) q = 0 which means p × 0 = 0
(f) p = q = 0 which means 0 × 0 = 0
Similarly, if (x + 2)(x+ 3) = 0, then either x + 2 = 0 or x + 3 = 0.
Solving for x, gives either x = -2 or x = -3.
Thus, 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.