Questions: Write the quadratic equation whose roots are -2 and 5, and whose leading coefficient is 3. (Use the letter x to represent the variable.)

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 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 -2 and 5, and the leading coefficient is 3. We will substitute these values into the formula to find the quadratic equation.

We are given the roots of the quadratic equation as \( r_1 = -2 \) and \( r_2 = 5 \), and the leading coefficient as \( a = 3 \).

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: \[ 3(x - (-2))(x - 5) = 0 \] This simplifies to: \[ 3(x + 2)(x - 5) = 0 \]

Now, we expand the equation: \[ 3[(x + 2)(x - 5)] = 3[x^2 - 5x + 2x - 10] = 3[x^2 - 3x - 10] \] Distributing the leading coefficient: \[ 3x^2 - 9x - 30 = 0 \]

Final Answer