To factor the trinomial \(4x^2 - 16x + 15\), let's break it down step by step:

### Step 1: Identify the terms
We have:
\[
4x^2 - 16x + 15
\]
This is a quadratic expression in the form \(ax^2 + bx + c\), where:
- \(a = 4\)
- \(b = -16\)
- \(c = 15\)

### Step 2: Multiply \(a\) and \(c\)
We multiply the coefficient of \(x^2\) (which is 4) by the constant term (which is 15):
\[
4 \times 15 = 60
\]

### Step 3: Find two numbers that multiply to 60 and add to -16
We now need to find two numbers that multiply to 60 and add to \(-16\). These numbers are:
\[
-10 \quad \text{and} \quad -6
\]
since \(-10 \times -6 = 60\) and \(-10 + -6 = -16\).

### Step 4: Rewrite the middle term
We rewrite \(-16x\) as \(-10x - 6x\):
\[
4x^2 - 10x - 6x + 15
\]

### Step 5: Factor by grouping
Group the terms in pairs and factor each group:
\[
(4x^2 - 10x) - (6x - 15)
\]
Now factor out the greatest common factor (GCF) from each group:
\[
2x(2x - 5) - 3(2x - 5)
\]

### Step 6: Factor out the common binomial
We now have a common binomial factor, \(2x - 5\):
\[
(2x - 5)(2x - 3)
\]

### Final Answer:
The factored form of \(4x^2 - 16x + 15\) is:
\[
(2x - 5)(2x - 3)
\]

Would you like more details on any of the steps or have any questions?

Here are 5 related questions for further practice:
1. How would you factor \(6x^2 - 13x + 5\)?
2. What happens when the trinomial cannot be factored into real numbers?
3. Can you solve \(4x^2 - 16x + 15 = 0\) using the quadratic formula?
4. How do you factor trinomials when the leading coefficient is 1?
5. What are the roots of \(4x^2 - 16x + 15 = 0\) based on the factorization?

**Tip:** When factoring, always check if the leading coefficient is not 1, as you will need to find pairs of numbers that both multiply and add to different values.

This page provides a detailed, step-by-step guide to factoring the quadratic trinomial 4x^2 - 16x + 15. Learn how to apply the factoring by grouping method, find two numbers that multiply to 60 and add to -16, and solve for the factors. Suitable for students in Grades 9-10.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation