Sum and Product of Roots
IB Mathematics AA · SL & HL · Shadow Worksheet
Practice
Question 1
For each quadratic, state the sum and product of the roots without solving:
(a)\(x^2-5x+6=0\)
(b)\(3x^2+7x-2=0\)
(c)\(x^2+px+q=0\)
(a)
Answer
Sum \(=5\); Product \(=6\)
(b)
Answer
Sum \(=-\tfrac{7}{3}\); Product \(=-\tfrac{2}{3}\)
(c)
Answer
Sum \(=-p\); Product \(=q\)
Question 2
The roots of \(x^2-6x+k=0\) are \(\alpha\) and \(\beta\). Given that \(\alpha^2+\beta^2=20\), find \(k\).
Use identity
\(\alpha^2+\beta^2=(\alpha+\beta)^2-2\alpha\beta=36-2k=20\)
Question 3
Write down a quadratic equation with integer coefficients whose roots are:
(a)\(3+\sqrt{2}\) and \(3-\sqrt{2}\)
(b)\(\dfrac{1}{2}\) and \(-3\)
(a)
Sum and product
Sum \(=6\); Product \(=9-2=7\)
(b)
Sum and product
Sum \(=-\tfrac{5}{2}\); Product \(=-\tfrac{3}{2}\)
Question 4
The roots of \(2x^2-5x+1=0\) are \(\alpha\) and \(\beta\). Without solving, find:
(a)\(\alpha+\beta\)
(b)\(\alpha\beta\)
(c)\(\dfrac{1}{\alpha}+\dfrac{1}{\beta}\)
(d)\(\alpha^2+\beta^2\)
(e)\((\alpha-\beta)^2\)
(c)
Simplify
\(\dfrac{\alpha+\beta}{\alpha\beta}=\dfrac{5/2}{1/2}\)
(d)
Identity
\((\alpha+\beta)^2-2\alpha\beta=\tfrac{25}{4}-1\)
(e)
Identity
\((\alpha+\beta)^2-4\alpha\beta=\tfrac{25}{4}-2\)
Question 5
The roots of \(x^3-6x^2+11x-6=0\) are \(\alpha,\beta,\gamma\). Find \(\alpha+\beta+\gamma\), \(\alpha\beta+\beta\gamma+\gamma\alpha\), and \(\alpha\beta\gamma\). Hence verify that \(\alpha=1,\beta=2,\gamma=3\) are the roots.
By Vieta's for cubic \(x^3-px^2+qx-r\)
Sum \(=6\); sum of products \(=11\); product \(=6\)
Verify
\(1+2+3=6\) ✓; \(2+3+6=11\) ✓; \(6=6\) ✓
Confirmed
\(\alpha=1,\;\beta=2,\;\gamma=3\)
Generated by Mathski · mathski.io · IB Mathematics AA Shadow Worksheets