Questions: Write the quadratic equation whose roots are 6 and 1, and whose leading coefficient is 3. (Use the letter x to represent the variable.)

Transcript text: Write the quadratic equation whose roots are 6 and 1, and whose leading coefficient is 3. (Use the letter $x$ to represent the variable.)

Solution

Solution Steps

To write a quadratic equation given the roots and the leading coefficient, use the fact that if $ r_1 $ and $ r_2 $ are the roots, the equation can be expressed as $ a(x - r_1)(x - r_2) = 0 $, where $ a $ is the leading coefficient. Substitute the given roots and coefficient into this formula.

Step 1: Identify the Roots and Leading Coefficient

The roots of the quadratic equation are given as $ r_1 = 6 $ and $ r_2 = 1 $. The leading coefficient is $ a = 3 $.

Step 2: Use the Quadratic Formula

The general form of a quadratic equation with roots $ r_1 $ and $ r_2 $ and leading coefficient $ a $ is: \[ a(x - r_1)(x - r_2) = 0 \]

Step 3: Substitute the Values

Substitute the given values into the formula: \[ 3(x - 6)(x - 1) = 0 \]

Step 4: Expand the Equation

Expand the expression: \[ 3(x^2 - 7x + 6) = 0 \] \[ 3x^2 - 21x + 18 = 0 \]