The solutions to $2 x^2+5 x-12=0$ are $x=\frac{3}{2}$ and $x=$ ?
Answer (1)
The problem provides a quadratic equation 2 x 2 + 5 x − 12 = 0 and one solution x = 2 3 .
Use the property that the product of the roots of a quadratic equation a x 2 + b x + c = 0 is a c .
Calculate the product of the roots: x 1 "." x 2 = 2 − 12 = − 6 .
Find the other root by solving for x 2 : x 2 = 2 3 − 6 = − 4 . The final answer is − 4 .
Explanation
Problem Analysis We are given the quadratic equation 2 x 2 + 5 x − 12 = 0 and one of its solutions, x = 2 3 . Our goal is to find the other solution.
Using the Product of Roots Let the two solutions to the quadratic equation be x 1 and x 2 . We know that x 1 = 2 3 . The product of the roots of a quadratic equation a x 2 + b x + c = 0 is given by a c . In our case, a = 2 , b = 5 , and c = − 12 .
Calculating the Product Therefore, the product of the roots is x 1 "." x 2 = a c = 2 − 12 = − 6 .
Substituting the Known Root Now, we substitute the known root x 1 = 2 3 into the equation: 2 3 x 2 = − 6 .
Solving for the Unknown Root To solve for x 2 , we multiply both sides of the equation by 3 2 : x 2 = − 6 × 3 2 = − 4
Final Answer Thus, the other solution to the quadratic equation is x = − 4 .
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For instance, when calculating the trajectory of a projectile, you often encounter quadratic equations that describe the height of the object as a function of time. Knowing how to find the roots of these equations allows you to determine when the projectile will hit the ground or reach its maximum height. This skill is also applicable in economics, where quadratic functions can model cost and revenue curves, helping businesses find optimal production levels.
