Quadratic Equations

1. The roots of the equation mx² + nx + 1 = 0 are -1 and 3. Find the values of m and n, substitute these values into the equation, and arrange the equation tidily, i.e. with integer coefficients.

Can someone explain how m = -1/3 and n = 2/3?

(x+1)(x-3) = x^2 - 2x - 3 = 0
I get m = 1, n = -2 ??????

Divide your answer by -3 to match the constant then compare coefficients.

Do you always divide by the biggest number or?

What? No...

The reason for division by -3 is that then your constant is the same in both $mx^2+nx+1=0$ and $-\frac{1}{3}x^2+\frac{2}{3}x+1=0$ so then you can compare the coefficients directly. If you were to compare them before the division then your constant would tell you that -3=1 which is clearly non sense.

Easier way you could've done this question in simply plug in x=-1 and x=3 into $mx^2+nx+1=0$ then solve simultaneous equations.

What stupid mistake am I doing this time..

2. The equation 3x² + 8x - 1 = 0 has roots α and β. Without solving this equation, find a quadratic equation whose roots are 2α and 2β, and arrange the equation with integer coefficients.

αβ= - 1/3
α + β= -8/3

2(αβ) = -1/3 * 2 = -2/3
2(α + β) 2(-8/3) = -`16/3
x² - 16/3x - 2/3 = 0
3x² - 16x - 2 = 0

Answer is: 3x² + 16x - 4 = 0 ????????/

Two mistakes. $2\alpha \times 2 \beta \neq 2 \alpha \beta$ and the coefficient of x should be $-\frac{b}{a}$.

How do I find alpha and beta? The course on cloudlearn just leaves you in the dark...