Questions: Write the quadratic equation whose roots are 5 and -3, and whose leading coefficient is 5. (Use the letter x to represent the variable.)

Transcript text: Write the quadratic equation whose roots are 5 and -3, and whose leading coefficient is 5. (Use the letter $x$ to represent the variable.)

Solution

Solution Steps

To write a quadratic equation given its roots and leading coefficient, we can use the fact that if $ r_1 $ and $ r_2 $ are the roots of the quadratic equation, then the equation can be expressed as $ a(x - r_1)(x - r_2) = 0 $, where $ a $ is the leading coefficient. In this case, the roots are 5 and -3, and the leading coefficient is 5. We substitute these values into the formula to get the quadratic equation.

Step 1: Identify the Roots and Leading Coefficient

We are given the roots of the quadratic equation as $ r_1 = 5 $ and $ r_2 = -3 $. The leading coefficient is $ a = 5 $.

Step 2: Construct the Quadratic Equation

Using the formula for a quadratic equation based on its roots, we have: \[ a(x - r_1)(x - r_2) = 0 \] Substituting the values, we get: \[ 5(x - 5)(x + 3) = 0 \]

Step 3: Expand the Equation

Expanding the equation: \[ 5[(x - 5)(x + 3)] = 5[x^2 + 3x - 5x - 15] = 5[x^2 - 2x - 15] \] Thus, we have: \[ 5x^2 - 10x - 75 = 0 \]