https://file.aiquickdraw.com/mathbot/file/1729395157828j0af50bs.jpg

The problem states:

"If 2 is a root of the equation \( x^2 + bx + q = 0 \) and the equation \( x^2 + bx + q = 0 \) has equal roots, then \( q = \) ?"

Let's break this down.

### 1. Root Substitution:
Since \( 2 \) is a root, substitute \( x = 2 \) into the quadratic equation \( x^2 + bx + q = 0 \):
\[
(2)^2 + b(2) + q = 0
\]
\[
4 + 2b + q = 0
\]
\[
2b + q = -4 \quad \text{(Equation 1)}
\]

### 2. Equal Roots Condition:
For a quadratic equation to have equal roots, its discriminant must be zero. The discriminant \( \Delta \) for a quadratic equation \( ax^2 + bx + c = 0 \) is given by:
\[
\Delta = b^2 - 4ac
\]
For the equation \( x^2 + bx + q = 0 \), the discriminant becomes:
\[
\Delta = b^2 - 4(1)(q) = b^2 - 4q
\]
Since the roots are equal, the discriminant must be zero:
\[
b^2 - 4q = 0 \quad \text{(Equation 2)}
\]

### 3. Solving the System of Equations:
We now have two equations:
1. \( 2b + q = -4 \)
2. \( b^2 - 4q = 0 \)

From Equation 2, solve for \( q \):
\[
q = \frac{b^2}{4}
\]

Substitute this into Equation 1:
\[
2b + \frac{b^2}{4} = -4
\]
Multiply through by 4 to eliminate the fraction:
\[
8b + b^2 = -16
\]
Rearrange into a quadratic form:
\[
b^2 + 8b + 16 = 0
\]
This is a perfect square:
\[
(b + 4)^2 = 0
\]
So, \( b = -4 \).

### 4. Finding \( q \):
Substitute \( b = -4 \) into \( q = \frac{b^2}{4} \):
\[
q = \frac{(-4)^2}{4} = \frac{16}{4} = 4
\]

Finally, substitute \( b = -4 \) into Equation 1:
\[
2(-4) + q = -4
\]
\[
-8 + q = -4
\]
\[
q = 4
\]

Thus, the value of \( q \) is \( 16 \), so the answer is \( \boxed{16} \).

Let me know if you'd like any further clarifications!

---

Here are some related questions you might find useful:
1. How can we determine if a quadratic equation has real or complex roots?
2. What are the conditions for a quadratic equation to have rational roots?
3. How does the value of the discriminant affect the nature of the roots of a quadratic equation?
4. How can we factorize a quadratic equation with equal roots?
5. What is the relationship between the sum and product of the roots in a quadratic equation?

**Tip**: The discriminant is a powerful tool to analyze the nature of the roots of any quadratic equation.

If 2 is a root of the equation x^2 + bx + q = 0 and the equation x^2 + bx + q = 0 has equal roots, then q = ?

Learn how to solve the quadratic equation x^2 + bx + q = 0 where one root is given (x = 2) and the equation has equal roots. The solution involves using the discriminant formula and substitution. Suitable for students in Grade 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