https://file.aiquickdraw.com/mathbot/file/1726821434160hlrmzp5b.jpg

Let's break down and solve the problem step by step.

### (a) Express \( 2x^2 + px + q \) in the form \( a(x + b)^2 + c \).

This part requires completing the square.

1. Start with the quadratic expression:
   \[
   2x^2 + px + q
   \]

2. Factor out the coefficient of \( x^2 \) (which is 2 in this case) from the first two terms:
   \[
   2(x^2 + \frac{p}{2}x) + q
   \]

3. To complete the square, take half the coefficient of \( x \), square it, and then add and subtract it inside the bracket:
   \[
   \frac{p}{2} \quad \text{(half of this is)} \quad \frac{p}{4}, \quad \left( \frac{p}{4} \right)^2 = \frac{p^2}{16}
   \]
   Add and subtract this inside the bracket:
   \[
   2 \left( x^2 + \frac{p}{2}x + \frac{p^2}{16} - \frac{p^2}{16} \right) + q
   \]
   Simplify:
   \[
   2 \left( \left( x + \

Express 2x^2 + px + q in the form a(x + b)^2 + c. The curve y = 2x^2 + px + q has a minimum point at (3, -2). Find the values of p and q.

Learn how to express a quadratic function in vertex form and find the values of p and q for a curve with a known minimum point. This solution involves completing the square and identifying key properties of parabolas. Suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation