Find two numbers that have a sum of 1 and a product of -12.
Answer (2)
Define the two numbers as x and y , and express the given conditions as equations: x + y = 1 and x y = − 12 .
Express y in terms of x : y = 1 − x .
Substitute into the product equation to get a quadratic equation: x ( 1 − x ) = − 12 , which simplifies to x 2 − x − 12 = 0 .
Solve the quadratic equation by factoring: ( x − 4 ) ( x + 3 ) = 0 , giving x = 4 or x = − 3 . Thus, the two numbers are 4 , − 3 .
Explanation
Problem Analysis We are asked to find two numbers that satisfy two conditions: their sum is 1 and their product is -12. We will use a quadratic equation to solve this problem.
Setting up Equations Let the two numbers be x and y . According to the problem, we have the following equations:
x + y = 1 (sum of the two numbers is 1)
x y = − 12 (product of the two numbers is -12)
Expressing y in terms of x From the first equation, we can express y in terms of x :
y = 1 − x
Substitution Substitute this expression for y into the second equation:
x ( 1 − x ) = − 12
Forming the Quadratic Equation Expand and rearrange the equation to form a quadratic equation in x :
x − x 2 = − 12 ⇒ x 2 − x − 12 = 0
Factoring the Quadratic Equation Now, we solve the quadratic equation x 2 − x − 12 = 0 . We can solve this by factoring. We look for two numbers that multiply to -12 and add to -1. These numbers are -4 and 3.
So, we can factor the quadratic equation as follows:
( x − 4 ) ( x + 3 ) = 0
Solving for x Setting each factor equal to zero gives us the possible values for x :
x − 4 = 0 ⇒ x = 4
x + 3 = 0 ⇒ x = − 3
Solving for y For each value of x , we find the corresponding value of y using y = 1 − x .
If x = 4 , then y = 1 − 4 = − 3 .
If x = − 3 , then y = 1 − ( − 3 ) = 4 .
Final Answer Thus, the two numbers are 4 and -3.
Examples
Understanding quadratic equations is crucial in various fields, such as physics and engineering. For instance, when designing a bridge, engineers use quadratic equations to model the parabolic shape of the bridge's arch. This ensures the bridge can withstand the forces acting upon it. Similarly, in physics, projectile motion, like the trajectory of a ball thrown in the air, can be described using quadratic equations. By understanding the roots and properties of these equations, we can predict the range and maximum height of the projectile, which is essential in sports and military applications.
The two numbers that have a sum of 1 and a product of -12 are 4 and -3. We find these by defining the numbers as variables, setting up equations for their sum and product, and solving a quadratic equation. The numbers can be verified by checking both the sum and the product: 4 + (-3) = 1 and 4 * (-3) = -12.
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