Write the quadratic equation whose roots are 3 and -4 , and whose leading coefficient is 4 . (Use the letter \( x \) to represent the variable.) П\( =0 \)

To construct a quadratic equation from the roots 3 and -4, we start with the factored form: \( (x - 3)(x + 4) \). By expanding this, we get \( x^2 + x - 12 \). Since we want the leading coefficient to be 4, we multiply the entire equation by 4, resulting in \( 4(x^2 + x - 12) = 4x^2 + 4x - 48 \). Therefore, the quadratic equation is: \[ 4x^2 + 4x - 48 = 0 \] --- If you're interested in the properties of quadratic equations, remember that the sum and product of roots can help you verify your answers. The sum, \(3 + (-4) = -1\), relates to the coefficient of \(x\) (which should be -1 when divided by the leading coefficient) and the product \(3 \cdot (-4) = -12\) corresponds to the constant term -12 divided by 4. In real-world applications, quadratic equations pop up in various fields! For example, they're essential in physics for calculating projectile motion, where the height of an object over time can be modeled by a quadratic equation. Whether you’re launching a basketball or planning a roller coaster, quadratics have you covered!