Find the quadratic equation whose roots are twice the roots of the quadratic equation 4x2 – 16x + 15 = 0.

1. Break Down the Problem

To find the new quadratic equation whose roots are twice the roots of the given quadratic equation $4x^2 - 16x + 15 = 0$, we will:

1. First, find the roots of the given equation.
2. Then, multiply these roots by 2.
3. Finally, construct the new quadratic equation using these new roots.

2. Relevant Concepts

A quadratic equation is generally expressed in the form: $ax^2 + bx + c = 0$ The roots of a quadratic equation $ax^2 + bx + c = 0$ can be found using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

3. Analysis and Detail

Step 1: Find the Roots of the Given Equation

For the equation $4x^2 - 16x + 15 = 0$:

• $a = 4$
• $b = -16$
• $c = 15$

Using the quadratic formula: $x = \frac{-(-16) \pm \sqrt{(-16)^2 - 4 \cdot 4 \cdot 15}}{2 \cdot 4}$ Calculating the discriminant: $(-16)^2 - 4 \cdot 4 \cdot 15 = 256 - 240 = 16$ Thus, $x = \frac{16 \pm \sqrt{16}}{8} = \frac{16 \pm 4}{8}$ Calculating the two roots: $x_1 = \frac{20}{8} = 2.5 \quad \text{and} \quad x_2 = \frac{12}{8} = 1.5$

Step 2: New Roots

The new roots are: $2x_1 = 2 \cdot 2.5 = 5 \quad \text{and} \quad 2x_2 = 2 \cdot 1.5 = 3$

Step 3: Construct the New Quadratic Equation

The new quadratic equation with roots $5$ and $3$ can be represented as: $(x - 5)(x - 3) = 0$ Expanding this: $x^2 - 8x + 15 = 0$

4. Verify and Summarize

The derived quadratic equation is $x^2 - 8x + 15 = 0$.

Final Answer

The quadratic equation whose roots are twice the roots of the quadratic equation $4x^2 - 16x + 15 = 0$ is: $x^2 - 8x + 15 = 0$