Question

write the quadratic equation whose roots are -4 and 1, and whose leading coefficient is 2. (use the letter x to represent the variable.)

Explanation:

Step1: Recall the factored form of a quadratic equation

If the roots of a quadratic equation are \( r_1 \) and \( r_2 \), the factored form is \( a(x - r_1)(x - r_2)=0 \), where \( a \) is the leading coefficient. Here, \( r_1=-4 \), \( r_2 = 1 \), and \( a = 2 \).
So the equation in factored form is \( 2(x - (-4))(x - 1)=0 \), which simplifies to \( 2(x + 4)(x - 1)=0 \).

Step2: Expand the factored form

First, multiply \( (x + 4)(x - 1) \):
\[

$$\begin{align*} (x + 4)(x - 1)&=x(x - 1)+4(x - 1)\\ &=x^2 - x+4x - 4\\ &=x^2 + 3x - 4 \end{align*}$$

\]
Then multiply by 2: \( 2(x^2 + 3x - 4)=2x^2+6x - 8 \)

Step3: Set the equation to zero

The quadratic equation is \( 2x^2 + 6x - 8 = 0 \)

Answer:

\( 2x^2 + 6x - 8 = 0 \)