We know that one root of the quadratic equation

$4x^2 + bx + c = 0$

is $x = 0.5$ and the other root is equal to $c$.

Step 1: Sum and Product of Roots

For a quadratic equation of the form $ax^2 + bx + c = 0$, if the roots are $r_1$ and $r_2$, then:

1. The sum of the roots is $r_1 + r_2 = -\frac{b}{a}$.
2. The product of the roots is $r_1 \cdot r_2 = \frac{c}{a}$.

In this problem:

• $a = 4$
• $r_1 = 0.5$
• $r_2 = c$

Step 2: Using the Sum of the Roots

Since $r_1 + r_2 = -\frac{b}{4}$, we have:

$0.5 + c = -\frac{b}{4}$

This gives us the equation:

$b = -4(0.5 + c) = -2 - 4c$

Step 3: Using the Product of the Roots

The product of the roots $r_1 \cdot r_2 = \frac{c}{4}$:

$0.5 \cdot c = \frac{c}{4}$

Solving this equation:

$0.5c = \frac{c}{4}$

Multiplying both sides by 4 to clear the fraction:

$2c = c$

This implies $c = 0$.

Step 4: Substitute $c = 0$ to Find $b$

Substituting $c = 0$ into our equation for $b$:

$b = -2 - 4(0) = -2$

Conclusion

The values of $b$ and $c$ are: $b = -2, \quad c = 0$

Would you like further details or have any questions?

1. How do you find the roots of a quadratic equation?
2. What if one of the roots was different? How would it change the solution?
3. How can you solve a quadratic equation without knowing any roots?
4. What are the applications of finding roots in real-world problems?
5. How does the quadratic formula relate to this method?

Tip: In quadratic equations, if one root is known, the sum and product of roots are powerful tools to find the other values.