Quadratic Equations SS1 Mathematics Lesson Note

A quadratic equation contains an equal sign and an unknown raised to the power 2. For example: 

1. a) 2×2 – 5x – 3 = 0
2. b) n2 + 50 = 27n 
3. c) 0 = (4a – 9)(2a + 1)
4. d) 49 = k2

SOLVING QUADRATIC EQUATIONS 

One way of solving a quadratic equation is to apply the following argument to a quadratic expression that has been factored. 

If the product of two numbers is 0, then one of the numbers (or possibly both of them) must be 0. For example, 

3  × 0 = 0, 0  × 5 = 0 and 0  0 = 0 

In general, if a × b = 0

    Then either a = 0 

        Or b = 0

        Or both a and b are 0 

 Example 1 

Solve the equation: (x – 2)(x + 7) = 0.

If (x – 2)(x + 7) = 0

Then either x – 2 = 0 or x + 7 = 0 

        x = 2 or -7 

 Example 2 

Solve the equation: d(d – 4)(d + 62) = 0.

(3a + 2)(2a – 7) = 0, then any one of the four factors of the LHS maybe 0, 

i.e d = 0 or d – 4 = 0 or d + 6 = 0 twice. 

⟹ d = 0, 4 or -6 twice. 

Solving quadratic equations using factorization method

The LHS of the quadratic equation m2 – 5m – 14 = 0 factorizes to give (m + 2)(m – 7) = 0. 

 Example 1 

Solve the equation 4y2 + 5y – 21 = 0 

4y2 + 5y – 21 = 0

⟹  (y + 3)(4y – 7)  = 0 

⟹either y + 3 = 0         or     4y – 7 = 0

    y = – 3     or     4y = 7 

    y = – 3    or    y = 7/4

    y = -3    or    134

check: by substitution: 

if y = -3 

4y2 + 5y – 21 = 36 – 15 – 21 = 0 

If y = 134, 

    4y2 + 5y – 21 = 4 x 7/4 x 7/4 + 5 x 7/4 – 21 

        = 494+ 354 – 21 = 0 

Example 2 

Solve the equation m2 = 16

Rearrange the equation. 

If m2 = 16

Then m2 – 16 = 0 

Factorise (difference of two squares)

(m – 4)(m + 4) = 0 

Either     m – 4 = 0    or     m + 4 = 0

    m = +4    or     m = -4

    m = 4 

ASSIGNMENT  

Solve the following equations. Check the results by substitution. 

1. (4b – 12)(b – 5) = 0    A. ½, 4      B. 3, 5       C. 4, 6        D.5, 3
2. (11 – 4x)2 = 0     A.113, 3      B.234, 3       C. 234 twice           D. 243 twice
3. (d – 5)(3d – 2) = 0    A. 5,23      B. 4, 5     C. 5, 9          D. 23, 5

Solve the following quadratic equations 

4. u2 – 8u – 9 = 0 A. – 9, 1      B. -1, 9    C. 1, 8          D. 9, -1

5. c2 = 25  A. 5     B. -5     C.+5          D.5

 Solve the equation 

6. 2×2 = 3x + 5

7. a2 – 3a = 0 

8. p2 + 7p + 12 = 0