Math Problem Statement
what two numbers equals -48 when multiplied and 13 when added
Solution
To find two numbers that multiply to and add up to , we can use algebra. Let the two numbers be and . We have the following system of equations:
We can solve this by substitution or by factoring. Let’s solve it step by step.
Step 1: Express one variable in terms of the other
From the second equation , we can express as:
Step 2: Substitute into the first equation
Now, substitute into the first equation :
Step 3: Expand and simplify
Distribute on the left side:
Rearrange this into a standard quadratic equation form:
Step 4: Solve the quadratic equation
Now, we’ll solve the quadratic equation . You can factor it, or use the quadratic formula. Let's try factoring:
We need two numbers that multiply to and add up to . These numbers are and , because:
So, the factored form is:
Step 5: Find the solutions
Set each factor equal to zero:
Step 6: Find the corresponding values for
Substitute these values of into :
- If , then .
- If , then .
Thus, the two numbers are 16 and -3.
Final Answer:
The two numbers are 16 and -3, which multiply to and add up to .
Would you like a more detailed explanation, or do you have any other questions?
Here are some related questions to explore:
- How can you solve a quadratic equation using the quadratic formula?
- What happens when the two numbers are both negative or both positive in such problems?
- How do you factor more complex quadratic equations?
- What is the discriminant in a quadratic equation, and how does it help to determine the number of real solutions?
- How can systems of equations be solved using matrices?
Tip: When dealing with quadratic equations, always check if you can factor the equation before applying the quadratic formula—it can save time!
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Math Problem Analysis
Mathematical Concepts
Algebra
System of Equations
Quadratic Equations
Formulas
x * y = -48
x + y = 13
Quadratic equation formula ax^2 + bx + c = 0
Theorems
Factoring quadratic equations
Solving systems of equations
Suitable Grade Level
Grades 8-10