Question

write the quadratic equation whose roots are -4 and 6, and whose leading coefficient is 1. (use the letter x to represent the variable.) \\(\square = 0\\)

Explanation:

Step1: Recall the factored form of a quadratic equation

If a quadratic equation has roots \( r_1 \) and \( r_2 \), then its factored form is \( a(x - r_1)(x - r_2)=0 \), where \( a \) is the leading coefficient. Here, \( r_1=-4 \), \( r_2 = 6 \), and \( a = 1 \).

Step2: Substitute the roots and leading coefficient into the factored form

Substitute \( r_1=-4 \), \( r_2 = 6 \), and \( a = 1 \) into \( a(x - r_1)(x - r_2)=0 \). We get \( 1\times(x - (-4))(x - 6)=0 \), which simplifies to \( (x + 4)(x - 6)=0 \).

Step3: Expand the factored form

Expand \( (x + 4)(x - 6) \) using the FOIL method. \( x\times x=x^{2} \), \( x\times(-6)=-6x \), \( 4\times x = 4x \), \( 4\times(-6)=-24 \). Combine like terms: \( x^{2}-6x + 4x-24=x^{2}-2x - 24 \). So the quadratic equation is \( x^{2}-2x - 24 = 0 \).

Answer:

\( x^{2}-2x - 24 = 0 \)