THINKING QUESTION
Create a quadratic equation in the form ax² + bx + c = 0 that crosses through the points (2/3, 0), and (-4, 0). The variables a, b, and c must be integers (whole numbers).
I think you mean
y = a x^2 + b x + c
y = 0 at x=2/3 and at x=-4
(x-2/3)(x+4) = 0
(3x-2)(x+4) = 0
3 x^2 +10 x - 8 = 0
so y =3 x^2 + 10 x - 8
You are given 2 intercepts, so we could just write
a(x+4)(3x - 2) = 0
since a is a constant, a ≠ 0, we could divide both sides by a
(x+4)(3x-2) = 0
3x^2 + 10x - 8 = 0
To create a quadratic equation in the form ax² + bx + c = 0 that crosses through the points (2/3, 0) and (-4, 0), we can use a process called factoring.
First, let's find the solutions of the equation based on the given points. Since both points have a y-coordinate of 0, it means that the equation crosses the x-axis at those points. Therefore, the solutions of the equation are x = 2/3 and x = -4.
We can rewrite these solutions in terms of common denominators: x = 2/3 can be rewritten as x = 6/9, and x = -4 can be written as x = -36/9.
Now we can set up two binomial factors to represent the quadratic equation:
(x - 6/9)(x + 36/9) = 0
To simplify the expression, we can multiply each binomial term by 9 to eliminate the denominators:
(9x - 6)(9x + 36) = 0
Expanding the expression, we get:
81x² + 270x - 216 = 0
Now we have the quadratic equation in the desired form: ax² + bx + c = 0, where a, b, and c are integers. In this case, a = 81, b = 270, and c = -216.
Therefore, the quadratic equation that crosses through the points (2/3, 0) and (-4, 0) is 81x² + 270x - 216 = 0.