How do you write a quadratic equation in standard form for a given set of zeros?

The standard form is

ax^2 + bc + c = 0

The value of ax^2 + bx +c will be zero when
x = [-b +/- sqrt (b^2-4ac)]/2a

If you want to write a quadratic equation that is zero at given values x=d and x=e, first write it as
(x-d)(x-e) = 0
and then multiply it out for the standard form:
x^2 -(d+e)x + de = 0

say you had (4,0) and (-3,0) given as zeros of a quadratic function.

Then x-4 = 0
plus x+3 = 0
and
(x-4)(x+3) = 0
and
x^2 - x - 12 = 0

I suppose I could generalize that to zeros at a and b
x-a = 0
x-b = 0
so
(x-a)(x-b) = 0
x^2 -(a+b)x + a b = 0

To write a quadratic equation in standard form for a given set of zeros, follow these steps:

Step 1: Identify the zeros of the quadratic equation. Let's say the zeros are a and b.

Step 2: Use the zero product property, which states that if the product of two factors is zero, then at least one of the factors must be zero. Set up two separate equations, one for each zero:

(x - a) = 0 and (x - b) = 0

Step 3: Expand and simplify both equations:

(x - a) = 0 becomes x - a = 0
(x - b) = 0 becomes x - b = 0

Step 4: Multiply the two equations together by multiplying both sides:

(x - a)(x - b) = 0

Step 5: Expand and simplify the product using the FOIL method:

x^2 - ax - bx + ab = 0

Step 6: Combine like terms:

x^2 - (a + b)x + ab = 0

Step 7: The quadratic equation is now in standard form, where the coefficient of x^2 is positive. The final equation is:

x^2 - (a + b)x + ab = 0

To write a quadratic equation in standard form for a given set of zeros, you can use the fact that the zeros (also known as roots) of a quadratic equation are the values of 'x' that make the equation equal to zero.

Let's assume a quadratic equation with zeros at 'p' and 'q'. In standard form, the equation will look like this:

(x - p)(x - q) = 0

To derive the standard form, follow these steps:

Step 1: Expand the equation:
x^2 - qx - px + pq = 0

Step 2: Simplify:
x^2 - (p + q)x + pq = 0

Therefore, the quadratic equation in standard form with zeros 'p' and 'q' is:
x^2 - (p + q)x + pq = 0

Keep in mind that the values of 'p' and 'q' are the zeros of the quadratic equation, so when you substitute these values into the equation, it will evaluate to zero.

Remember, when working with quadratic equations, the standard form is typically written as ax^2 + bx + c = 0, where 'a', 'b', and 'c' are constants and 'x' is the variable.