Unit 2 - Functions
Required Prior Knowledge
Questions
Find the roots of the following equations:
a) \(x^{2}+5x+6=0 \)
b) \(x^{2}-3x-10=0\)
c) \(2x^{2}+5x+3=0\)
d) \(3x^{2}-5x-2=0\)
Solutions
Get Ready
Questions
Calculate the sum and the product of the roots of:
a) \(x^{2}+5x+6=0 \)
b) \(x^{2}-3x-10=0\)
c) \(2x^{2}+5x+3=0\)
d) \(3x^{2}-5x-2=0\)
What do you notice? What do you wonder?
Solutions
Notes
For any quadratic equation \(ax^{2}+bx+c=0\), with solutions \(x_{1}\) and \(x_{2}\) we have that $$x_{1}+x_{2}=-\frac{b}{a}$$and$$x_{1}\times x_{2}=\frac{c}{a}$$Compare this to what you found in the Get Ready.
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Suppose we have the quadratic equation \(ax^{2}+bx+c=0\) with roots \(\alpha\) and \(\beta\).
We know we can rewrite this equation in factorised form as$$a\left(x-\alpha\right)\left(x-\beta\right)$$Now expanding this we get$$a\left(x^{2}-\alpha x-\beta x+\alpha\beta\right)=0$$Which we can simplify to$$a\left(x^{2}-\left(\alpha+\beta\right)x+\alpha\beta\right)=0$$And then expand further to$$ax^{2}-a\left(\alpha+\beta\right)x+a\alpha\beta=0$$Since this is just a rearrangement of the original quadratic equation, we can equate the coefficient of \(x\) and the constant term to get$$b=-a\left(\alpha+\beta\right)\implies \alpha+\beta=-\frac{b}{a}\\c=a\alpha\beta\implies\alpha\beta=\frac{c}{a}$$
Examples and Your Turns
Example
Find the sum and product of the roots of $$25x^{2}-20x+1=0$$
Your Turn
Find the sum and product of the roots of $$4x^{2}-5x-1=0$$
Example
The quadratic equation \(3x^{2}+kx-7=0\) has roots \(\alpha\) and \(\beta\). If \(\alpha +\beta=4\), find the value of \(k\).
Your Turn
The quadratic equation \(2x^{2}-5x+k=0\) has roots \(\alpha\) and \(\beta\). If \(\alpha \beta=-3\), find the value of \(k\).
Your Turn
The quadratic equation \(ax^{2}+bx+12=0\) has roots \(\alpha\) and \(\beta\). Given that \(\alpha +\beta=-4\) and \(\alpha\beta=3\), find the values of \(a\) and \(b\).
Your Turn
The quadratic equation \(x^{2}+px+q=0\) has roots \(\alpha\) and \(\beta\). Given that \(\alpha +\beta=5\) and \(\alpha\beta=-6\), find the values of \(p\) and \(q\).
Your Turn
The roots of the quadratic equation \(3x^{2}-5x+2=0\) are \(x_{1}\) and \(x_{2}\).
Without solving the equation, find:
a) \(\frac{1}{x_{1}}+\frac{1}{x_{2}}\)
b) \(x_{1}^{2}+x_{2}^{2}\)
c) \(\frac{2}{x_{1}^{2}}+\frac{2}{x_{2}^{2}}\)
Your Turn
One root of the quadratic equation \(kx^{2}-10x+3=0\) is twice the other. Find the value of \(k\).
Your Turn
For the equation \(px^{2}+\left(p-3\right)x+4=0\), the product of the roots is twice the sum of the roots.
Find \(p\), and hence the two roots.
Your Turn
The quadratic equation \(3x^{2}+\left(m+1\right)x+\left(m-1\right)=0\) has roots \(\alpha\) and \(\beta\).
Given that one root is the reciprocal of the other, find the value of \(m\).
Your Turn
The quadratic equation \(x^{2}+\left(k+2\right)x+\left(2k+1\right)=0\), where \(k\ne -2\), has one root which is three less than the other.
Find \(k\).
Your Turn
The roots of the equation \(3x^{2}-4x-2=0\) are \(\alpha\) and \(\beta\).
a) Find a quadratic equation with roots \(2\alpha\) and \(2\beta\).
b) Find a quadratic equation with roots \(\frac{1}{\alpha}\) and \(\frac{1}{\beta}\).
Find a quadratic equation with roots \(\alpha^{2}\) and \(\beta^{2}\).
Notes
For the cubic equation \(ax^{3}+bx^{2}+cx+d=0\), with solutions \(x_{1}\), \(x_{2}\) and \(x_{3}\), we have some similar results:$$x_{1}+x_{2}+x_{3}=-\frac{b}{a}\\x_{1}\times x_{2}\times x_{3}=-\frac{d}{a}\\x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}=\frac{c}{a}$$
We can also generalise this to any polynomial function.
Consider \(f\left(x\right)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{2}x^{2}+a_{1}x+a_{0}\) with zeros \(x_{1},x_{2}, \cdots, x_{n}\).
Then$$x_{1}+x_{2}+\cdots+x_{n}=-\frac{a_{n-1}}{a_{n}}$$and$$x_{1}\times x_{2}\times \cdots\times x_{n}=\left(-1\right)^{n}\frac{a_{0}}{a_{n}}$$We can also generalise to the sum of groups of products (though this is not explicitly in the course).$$\sum \left(x_{i_{1}}\times x_{i_{2}}\times \cdots\times x_{i_{k}}\right)=\left(-1\right)^{k}\frac{a_{n-k}}{a_{n}}$$
In the formula booklet you are given only the general sum and product versions above, and have to use this for quadratics and cubics.
Examples and Your Turns
Example
Given that the roots of the cubic equation \(2x^{3}+4x^{2}-7x+5=0\) are \(x_{1}\), \(x_{2}\) and \(x_{3}\), without solving the equation, find:
a) \(x_{1}+x_{2}+x_{3}\)
b) \(x_{1}\times x_{2}\times x_{3}\)
c) \(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}\)
d) \(\frac{1}{x_{1}}+\frac{1}{x_{2}}+\frac{1}{x_{3}}\)
e) \(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}\)
Your Turn
Given that the roots of the cubic equation \(3x^{3}+5x^{2}+2x-4=0\) are \(x_{1}\), \(x_{2}\) and \(x_{3}\), without solving the equation, find:
a) \(x_{1}+x_{2}+x_{3}\)
b) \(x_{1}\times x_{2}\times x_{3}\)
c) \(\frac{1}{x_{1}x_{2}}+\frac{1}{x_{1}x_{3}}+\frac{1}{x_{2}x_{3}}\)
Your Turn
The roots of the equation \(x^{3}+px^{2}+qx+r=0\) are \(1\), \(2\) and \(-3\).
Find the values of \(p\), \(q\) and \(r\).
Your Turn
The roots of the equation \(x^{3}-7x^{2}+kx-13=0\) are \(r_{1}\), \(r_{2}\) and \(r_{3}\).
It is given that one of these roots is \(2+i\).
a) Find the other two roots.
b) Hence, or otherwise, find the value of \(k\).
Your Turn
A polynomial \(P\left(x\right)=x^{4}+ax^{3}+bx^{2}+cx+d\) has real coefficients.
Three of its roots are \(1\), \(2+i\) and \(-3\).
Determine the value of \(a\), \(b\), \(c\) and \(d\).
Key Facts
Use this applet to generate a prompt for a Key Fact that you need to know for the course. The idea is that you should KNOW these key facts in order to be able to solve problems.
Taking it Deeper
Conceptual Questions to Consider
If the sum of the roots of a quadratic is positive, what can you say about the coefficient of the \(x\) term?
Can a cubic polynomial with real coefficients have a positive product of roots if all its roots are real and negative? Explain.
Common Mistakes / Misconceptions
The biggest mistake is to get the sign wrong. Remember that the sign of the sum is ALWAYS negative. The sign of the product depends on the order of the polynomial. If the order of the polynomial is even, the product is positive; if the order of the polynomial is odd, the product is negative.
Misunderstanding the way the formula is given in the formula booklet is a common issue - make sure you are familiar with it!
When a term is missing, for example in \(x^{3}-4x+5=0\) there is no \(x^{2}\), not realising this means the coefficient is \(0\).
Connecting This to Other Skills
This skill builds upon Polynomials (2.19), The Factor Theorem (2.20) and The Conjugate Root Theorem (2.21), all of which might be required to solve these problems.
Often you will need to solve Simultaneous Equations (PK4) or Quadratic Equations (PK3).
This also build upon the ideas of Quadratic Equations with complex roots (2.12), as these rules work no matter the nature of the roots themselves.
Self-Reflection
What was the most challenging part of this skill for you?
What are you still unsure about that you need to review?