2.10.3 Quadratic Equations, SPM Practice (Paper 2)

2.10.3 Quadratic Equations, SPM Practice (Paper 2) Question 5: Given 3t and (t – 7) are the roots of the quadratic equation 4×2 – 4x + m = 0 where m is a constant. (a)  Find the values of t and m. (b)  Hence, form the quadratic equation with roots 4t and 2t + 6. … Read more

2.10.2 Quadratic Equations, SPM Practice (Paper 2)

2.10.2 Quadratic Equations, SPM Practice (Paper 2) Question 3: If α and β are the roots of the quadratic equation 3×2 + 2x– 5 = 0, form the quadratic equations that have the following roots. (a)  2 α  and  2 β (b)  ( α + 2 β )  and  ( β + 2 α ) … Read more

2.10.1 Quadratic Equations, SPM Practice (Paper 2)

2.10.1 Quadratic Equations, SPM Practice (Paper 2) Question 1: (a)  Find the values of k if the equation (1 – k) x2– 2(k + 5)x + k + 4 = 0 has real and equal roots. Hence, find the roots of the equation based on the values of k obtained. (b)  Given the curve y … Read more

2.9.3 Quadratic Equation, SPM Practice (Paper 1)

Question 11: The quadratic equation x 2 −4x−1=2p(x−5) , where p is a constant, has two equal roots. Calculate the possible values of p. Solution: Question 12: Find the range of values of k for which the equation x 2 −2kx+ k 2 +5k−6=0 has no real roots. Solution: Question 13: Find the range of … Read more

2.9.2 Quadratic Equation, SPM Practice (Paper 1)

Question 6: Write and simplify the equation whose roots are the reciprocals of the roots of 3 x 2 +2x−1=0 , without solving the given equation. Solution: Question 7: Find the value of p if one root of x 2 +px+8=0 is the square of the other. Solution: Question 8: If one root of 2 … Read more

2.9.1 Quadratic Equation, SPM Practice (Paper 1)

Question 1: Solve the following quadratic equations by factorisation. (a)  x 2 −5x−10=−4 (b) 3−x−2 x 2 =0 (c) 11a=2 a 2 +12 (d)  2x+7 3x−2 =x Solution: Question 2: Solve the following quadratic equations by completing the square. (a) 5 x 2 +10x−3=0 (b) 2 x 2 −5x−6=0 Solution: Question 3: Solve the following quadratic equations by using … Read more

2.8 Discriminant of a Quadratic Equation

The Discriminant The expression b 2 − 4 a c in the general formula is called the discriminant of the equation, as it determines the type of roots that the equation has. Example Determine the nature of the roots of the following equations. a. 5 x 2 − 7 x + 3 = 0 b. … Read more

2.7 Forming New Quadratic Equation given a Quadratic Equation

Example If the roots of x 2 − 3 x − 7 = 0   are α  and β  , find the equation whose roots are α 2 β   and α β 2 . Solution Part 1 : Find SoR and PoR for the quadratic equation in the question Part 2 : Form a … Read more

2.6.1 Sum of Roots and Product of Roots (Examples)

ExampleThe roots of 2 x 2 + 3 x − 1 = 0   are α and β.  Find the values of (a) ( α + 1 ) ( β + 1 ) (b) 1 α + 1 β (c) α 2 β + α β 2 (d) α β + β α [Clue: α 2 … Read more

2.6 Finding the Sum of Roots and Product of Roots

2.6 Finding the Sum of Roots (SoR) and Product of Roots (PoR) Example Find the sums and products of the roots of the following equations. a. x 2 + 7 x − 3 = 0 b. x ( x − 1 ) = 5 ( 1 − x ) Answer: (a) x 2 + 7 … Read more