Write the equation in standard form of the quadratic that has a root of (3,0) and a vertex at (1,-4), then list out the values of a, b and c.

a = 1, b = -2, c = -3

Step-by-step explanation:

The equation in factored form shows the roots with the "r" and "s" variables. Note that they are written negative "r" and negative "s" in the formula.

y = a(x - r)(x - s)

The vertex is always halfway between the two roots.

<[root][vertex][root]>

The vertex is at x = 1, and one root is at x = 3. The other root is at x = -1.

Write the equation in factored form to show the two roots.

y = a(x - 3)(x + 1)

Substitute the vertex (1, -4) into the equation. The point is written (x, y).

-4 = a(1 - 3)(1 + 1)

Solve for "a".

-4 = a(-2)(2)      Simplify within the brackets

-4 = -4a          Multiply -2 and 2. Divide both sides by -4

a = 1             Solved for "a" (and kept the variable on the left side).

Substitute a = 1 back into the formula.

y = a(x - 3)(x + 1)

y = 1(x - 3)(x + 1)

We don't have to write multiplied by 1 though, because anything multiplied by 1 is itself.

y = (x - 3)(x + 1)

Since we are looking for "a", "b", and "c", find the equation in standard form.

y = ax² + bx + c

To change factored form into standard form, expand the brackets.

y = (x - 3)(x + 1)

y = x² + x - 3x - 3

y = x² - 2x - 3

Compare the expanded equation to y = ax² + bx + c

∴ a = 1, b = -2, c = -3

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