Pointers to Review.pdf

1  Pointers to Review Grade 9 mathematics 2 Pointers to Review 1. Quadratic Equation 1.1 Definition 1.2 Standard Form A quadratic equation in one variable is a mathematical sentence of degree 2 that can be written in the following standar...

1  Pointers to Review Grade 9 mathematics 2 Pointers to Review 1. Quadratic Equation 1.1 Definition 1.2 Standard Form A quadratic equation in one variable is a mathematical sentence of degree 2 that can be written in the following standard form 𝑎𝑎𝑎𝑎 2 + 𝑏𝑏𝑏𝑏 + 𝑐𝑐 = 0 Where a, b, and c are real numbers and a ≠ 0 Pointers to Review 1.3 Methods of Solving 1.3.1. Extracting Square Roots Example: For equations like 𝑥𝑥 2 = 49, 𝑥𝑥 2 = 49, take the square root of both sides x = ±7 Pointers to Review 1.3 Methods of Solving 1.3.2 Factoring For example, 𝑥𝑥 2 -5x+6=0 can be factored into (x−2)(x−3)=0, giving solutions 𝑥𝑥1 = 2, 𝑥𝑥2 = 3 by applying Zero Product Property Pointers to Review 1.3 Methods of Solving 1.3.3. Completing the Square To complete the square of the expression x2 + bx, add the square of half the coefficient of x to make 𝑏𝑏 2 x2 + bx+( ). 2 Pointers to Review 1.3 Methods of Solving 1.3.3. Completing the Square For example, 𝑥𝑥 2 + 6𝑥𝑥 − 2 = 0 6 6 𝑥𝑥 2 + 6𝑥𝑥 + ( )2 = 2 +( )2 2 2 𝑥𝑥 2 + 6𝑥𝑥 + 9 = 2 + 9 (𝑥𝑥 + 3)2 = 11 (𝑥𝑥 + 3)2 = 11 x+3=± 11 x = -3± 11 𝑥𝑥1 = −3 + 11 𝑜𝑜𝑜𝑜 𝑥𝑥1 ≈ 0.32 𝑥𝑥2 = −3 − 11 or 𝑥𝑥2 ≈ −6.32 Pointers to Review 1.3 Methods of Solving 1.3.4. Quadratic Formula −𝒃𝒃± 𝒃𝒃𝟐𝟐 −𝟒𝟒𝟒𝟒𝟒𝟒 𝐱𝐱 = 𝟐𝟐𝟐𝟐 Step 1. Rewrite the equation into standard form. Step 2. Substitute the values of a, b, and c in your quadratic formula. Step 3. Simplify Pointers to Review For example, 𝒙𝒙𝟐𝟐 + 𝟑𝟑𝟑𝟑 − 𝟏𝟏𝟏𝟏 = 𝟎𝟎 a = 1, b=3, c= -10 By substitution; −𝒃𝒃± 𝒃𝒃𝟐𝟐 −𝟒𝟒𝟒𝟒𝟒𝟒 𝐱𝐱 = 𝟐𝟐𝟐𝟐 −𝟑𝟑 + 𝟕𝟕 𝑥𝑥1 = 𝟐𝟐 𝑥𝑥1 = 2 −𝟑𝟑−𝟕𝟕 𝑥𝑥2 = 𝟐𝟐 𝑥𝑥2 = 5 Pointers to Review 1.4 Discriminant and Nature of Roots The discriminant (d) of a quadratic equation is used to determine the nature of its roots. It is given by the formula 𝒅𝒅 = 𝒃𝒃 𝟐𝟐 − 𝟒𝟒𝟒𝟒𝟒𝟒 where 𝑎𝑎, 𝑏𝑏, and 𝑐𝑐 are the coefficients of the quadratic equation in standard form a𝑥𝑥 2 +𝑏𝑏𝑥𝑥+𝑐𝑐=0. Pointers to Review 1.4 Discriminant and Nature of Roots 𝟐𝟐 𝒅𝒅 = 𝒃𝒃 − 𝟒𝟒𝟒𝟒𝟒𝟒 Discriminant Nature of Roots d>0 Two distinct real roots d=0 One real root of multiplicity two d