Math Problem Statement

what two numbers equals -48 when multiplied and 13 when added

Solution

To find two numbers that multiply to 48-48 and add up to 1313, we can use algebra. Let the two numbers be xx and yy. We have the following system of equations:

xy=48(multiplication)x \cdot y = -48 \quad \text{(multiplication)} x+y=13(addition)x + y = 13 \quad \text{(addition)}

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 x+y=13x + y = 13, we can express yy as: y=13xy = 13 - x

Step 2: Substitute into the first equation

Now, substitute y=13xy = 13 - x into the first equation xy=48x \cdot y = -48: x(13x)=48x \cdot (13 - x) = -48

Step 3: Expand and simplify

Distribute xx on the left side: 13xx2=4813x - x^2 = -48

Rearrange this into a standard quadratic equation form: x213x48=0x^2 - 13x - 48 = 0

Step 4: Solve the quadratic equation

Now, we’ll solve the quadratic equation x213x48=0x^2 - 13x - 48 = 0. You can factor it, or use the quadratic formula. Let's try factoring:

We need two numbers that multiply to 48-48 and add up to 13-13. These numbers are 16-16 and 33, because: 163=48and16+3=13-16 \cdot 3 = -48 \quad \text{and} \quad -16 + 3 = -13

So, the factored form is: (x16)(x+3)=0(x - 16)(x + 3) = 0

Step 5: Find the solutions

Set each factor equal to zero: x16=0x=16x - 16 = 0 \quad \Rightarrow \quad x = 16 x+3=0x=3x + 3 = 0 \quad \Rightarrow \quad x = -3

Step 6: Find the corresponding values for yy

Substitute these values of xx into y=13xy = 13 - x:

  • If x=16x = 16, then y=1316=3y = 13 - 16 = -3.
  • If x=3x = -3, then y=13(3)=16y = 13 - (-3) = 16.

Thus, the two numbers are 16 and -3.

Final Answer:

The two numbers are 16 and -3, which multiply to 48-48 and add up to 1313.

Would you like a more detailed explanation, or do you have any other questions?

Here are some related questions to explore:

  1. How can you solve a quadratic equation using the quadratic formula?
  2. What happens when the two numbers are both negative or both positive in such problems?
  3. How do you factor more complex quadratic equations?
  4. What is the discriminant in a quadratic equation, and how does it help to determine the number of real solutions?
  5. 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