Transcript Slide 1

Investigate the relationship
between the roots of a cubic
equation and its coefficients.
Sum of roots
Product of roots
What about polynomial
equations of degree n?
Coefficients
1) The roots of the quadratic equation 𝑥 2 − 3𝑥 + 5 = 0 are α and β.
(a) Write down the value of α + β and the value of αβ.
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a
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c
a
 2   2       2
2
 3   3       3    
3
Prove the quadratic
formula by completing the
square on ax2 + bx + c = 0
(b) Without solving the quadratic equation, find the value of α² + β².
Hence explain why α and β cannot both be real.
(c) Show that α³ + β³ = -18.
2)
The roots of the quadratic equation 𝑥 2 + 4𝑥 − 3 = 0 are α and β.
(a) Without solving the equation, find the value of:
2
2
(i) α² + β²
(ii) 𝛼 2 +
𝛽2 +
𝛽
Use the quadratic formula
to prove the following:
𝛼
(b) Determine a quadratic equation with integer coefficients which has roots
2
2
𝛼 2 + 𝑎𝑛𝑑 𝛽 2 +
𝛽
3)
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a
𝛼
The roots of the quadratic equation 𝑥 2 + 7 + 𝑝 𝑥 + 𝑝 = 0 are α and β.
(a) Write down the value of α + β and the value of αβ in terms of p.
(b)
Find the value of α² + β² in terms of p.
(c)
(i) Show that (α - β)² = p² + 10p + 49.
(ii) Given that α and β differ by 5, find the possible values of p.
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