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SS2: MATHEMATICS - 2ND TERM

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  1. Sequence | Week 1
    2 Topics
  2. Series | Week 2
    2 Topics
    |
    1 Quiz
  3. Geometric Progression | Week 3
    2 Topics
    |
    1 Quiz
  4. Linear Equations & Formulae | Week 4
    5 Topics
    |
    1 Quiz
  5. Quadratic Equations II | Week 5
    2 Topics
  6. Quadratic Equations III | Week 6
    1 Topic
  7. Quadratic Equations IV | Week 7
    3 Topics
    |
    1 Quiz
  8. Simultaneous Equations I | Week 8
    2 Topics
  9. Simultaneous Equations II | Week 9
    2 Topics
    |
    1 Quiz
  10. Algebraic Fractions | Week 10
    5 Topics
    |
    1 Quiz



Lesson 6, Topic 1
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Finding a Quadratic Equation whose Roots are Given

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Topic Content:

  • Finding a Quadratic Equation whose Roots are Given

If α and β are the roots of a quadratic equation, then either x = α or x = β

i.e. x – α = 0 or x – β = 0

∴ (x – α)(x – β) = 0

By expansion we have

\( \scriptsize x^2 \: – \: αx \: – \: xβ\: + \: αβ = 0\)

= \( \scriptsize x^2 \: -\: \left(\: α \: + \: β \right)x \: + \: αβ = 0\)

Note that: 

(i) The coefficient of x2 is 1

(ii) α + β = Sum of the roots.

Observe that it is the coefficient of x with the sign changed

(iii) αβ = Product of roots. Observe that it is the constant term

Thus we have:

\( \scriptsize x^2 \: – \: \left(\: α \: + \: β \right)x \: + \: αβ = \scriptsize x^2\: +\: \normalsize \frac{b}{a}\scriptsize x\: +\: \normalsize \frac{c}{a}\)

Comparing coefficients, we have

\( \scriptsize \:- \left(\: α \: + \: β \right) = \normalsize \frac{b}{a} \scriptsize \: and \: αβ = \normalsize \frac{c}{a} \)

\( \scriptsize \left(\: α \: + \: β \right) = \: – \normalsize \frac{b}{a} \scriptsize \: and \: αβ = \normalsize \frac{c}{a} \)

Example 6.1.1:

Find:

(i) a quadratic equation whose roots are 5 & -4, \( \frac{2}{5}\) and \( \scriptsize -1 \normalsize \frac{1}{2} \)

(ii) the sum and product of roots of 15y2 – 17y + 5 = 0 and 6x2 = 2x + 20

(iii) If α = \( – \frac{1}{2}\) is a root of the quadratic  equation 8x2 – bx – 3 = 0, find the value of

(a) b

(b) the other root, β  

(c) \( \left (\frac{1}{α} \: – \: \frac{1}{β} \right)^2 \)

(iv) If one root of the quadratic equation 6x2 – kx + 48 = 0 is the square of the other, find the two roots and hence find the value of k.

(i) a quadratic equation whose roots are 5 & -4, \( \frac{2}{5}\) and \( \scriptsize -1 \normalsize \frac{1}{2} \)

Solution

a. Let x = 5

 

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