Roots of a Quadratic Equation
The big rules! (Quadratic)
- \(\alpha+\beta=-\frac{b}{a}\)
- \(\alpha\beta=\frac{c}{a}\)
Expanding a quadratic
- \(ax^{2}+bx+c\equiv a(x-\alpha)(x-\beta)\)
- \(a(x^2-(\alpha+\beta)x+\alpha\beta)\)
- \(ax^2-a(\alpha+\beta)x+a\alpha\beta\)
- \(\alpha+\beta = -\frac{b}{a}\)
- \(\alpha\beta=\frac{c}{a}\)
If \(\alpha\) and \(\beta\) are roots of the equation \(ax^2+bx+c=0\) then: - Sum of roots: \(\alpha+\beta=-\frac{b}{a}\)
- Product of roots: \(\alpha\beta = \frac{c}{a}\)
The roots of \(ax^{2}+bx+c=0\) are \(\alpha = -\frac{3}{2}\) and \(\beta=\frac{5}{4}\). Find integer values of \(a, b,c\).
- \(\alpha+\beta=-\frac{3}{2}+\frac{5}{4} =-\frac{1}{4}=\frac{b}{a} \to 4b=a\)
- \(\alpha\beta=\frac{c}{a}=-\frac{3}{2}\times\frac{5}{4}=-\frac{15}{8}\)
- We set \(a=1\) because we are finding a solution to this. (There are more than one)
- \(x^2+\frac{1}{4}x-\frac{15}{8}\)
- Times through by 8: \(8x^2+2x-15\)
- Done
Q7 (Complex): One of the roots of a quadratic is \(\alpha=-1-4i\).
- If \(\alpha=-1-4i\) then \(\beta=-1+4i\)
- \(\alpha+\beta=-2 = -\frac{b}{a} \to a=2b\)
- \(\alpha\beta=17=\frac{c}{a}\)
- \(x^2+2x+17=0\)
Q8 (Values of k): \(kx^2+(k-3)x-2=0\) - find k if the sum of the roots is \(4\)
- \(3-k=4\)
- \(k=-1\)