Question

A quadratic function is given.

f(x) = 2x^2 + 4x + 3
(a) Express the quadratic function in standard form.

Answer 1

To express the quadratic function f(x) = 2x^2 + 4x + 3 in standard form, complete the square to get f(x) = 2(x + 1)^2 - 5, which reveals the vertex of the parabola at (-1, -5).

Step-by-step explanation:

Expressing a Quadratic Function in Standard Form

To express the quadratic function f(x) = 2x^2 + 4x + 3 in standard form, also known as vertex form, we need to complete the square. The standard form of a quadratic function is typically written as f(x) = a(x-h)^2 + k, where (h, k) is the vertex of the parabola formed by the graph of the quadratic equation.

Here's how we can rewrite the given quadratic function:

1. Divide the coefficient of the x term by 2, which is 4/2 = 2, and then square it to get 4.
2. Add and subtract this value inside the parentheses to complete the square: f(x) = 2(x^2 + 2x + 4 - 4) + 3.
3. Rewrite the equation by grouping the perfect square trinomial and combining the constants: f(x) = 2((x + 1)^2 - 4) + 3.
4. Finally, distribute the coefficient and combine like terms to complete the process: f(x) = 2(x + 1)^2 - 5.

In this form, it is clear that the vertex of the parabola is at (-1, -5).

Answer 2

2x² + 4x + 3 = 0

Step-by-step explanation:

The function is said to be quadratic if it has highest degree = 2.

Further, The standard form of Quadratic Equation is:

ax² + bx + c = 0

where, a ≠ 0

a, b and c are constants

and x is unknown variable.

Thus, The Standard form of given Quadratic Equation is 2x² + 4x + 3 = 0