What 2 numbers multiply to 30 but add to 16

What 2 numbers multiply to 30 but add to 16

The Correct Answer and Explanation is:

Let’s break this down step-by-step. You are looking for two numbers that multiply to 30 and add to 16.

To solve this, let’s denote the two numbers as x and y.

1. Multiplication condition:
   x×y=30x \times y = 30x×y=30
2. Addition condition:
   x+y=16x + y = 16x+y=16

Step 1: Express one variable in terms of the other

From the second equation x+y=16x + y = 16x+y=16, solve for one of the variables. Let’s solve for xxx:x=16−yx = 16 – yx=16−y

Step 2: Substitute into the multiplication equation

Now, substitute x=16−yx = 16 – yx=16−y into the first equation x×y=30x \times y = 30x×y=30:(16−y)×y=30(16 – y) \times y = 30(16−y)×y=30

Step 3: Expand and simplify

Now expand the equation:16y−y2=3016y – y^2 = 3016y−y2=30

Rearrange this into a standard quadratic form:y2−16y+30=0y^2 – 16y + 30 = 0y2−16y+30=0

Step 4: Solve the quadratic equation

Now, we can solve this quadratic equation using the quadratic formula:y=−(−16)±(−16)2−4(1)(30)2(1)y = \frac{-(-16) \pm \sqrt{(-16)^2 – 4(1)(30)}}{2(1)}y=2(1)−(−16)±(−16)2−4(1)(30)​​

Simplify the terms inside the square root:y=16±256−1202y = \frac{16 \pm \sqrt{256 – 120}}{2}y=216±256−120​​y=16±1362y = \frac{16 \pm \sqrt{136}}{2}y=216±136​​y=16±11.662y = \frac{16 \pm 11.66}{2}y=216±11.66​

So, the two possible solutions for yyy are:y=16+11.662≈13.83y = \frac{16 + 11.66}{2} \approx 13.83y=216+11.66​≈13.83

ory=16−11.662≈2.17y = \frac{16 – 11.66}{2} \approx 2.17y=216−11.66​≈2.17

Step 5: Find corresponding xxx values

Using the equation x=16−yx = 16 – yx=16−y, we can now find the corresponding xxx values:

For y≈13.83y \approx 13.83y≈13.83,x≈16−13.83=2.17x \approx 16 – 13.83 = 2.17x≈16−13.83=2.17

For y≈2.17y \approx 2.17y≈2.17,x≈16−2.17=13.83x \approx 16 – 2.17 = 13.83x≈16−2.17=13.83

Conclusion:

So, the two numbers are approximately 2.17 and 13.83. These values satisfy both the conditions:

• Their product is close to 2.17×13.83≈302.17 \times 13.83 \approx 302.17×13.83≈30,
• Their sum is 2.17+13.83=162.17 + 13.83 = 162.17+13.83=16.

These numbers are not exact integers, but they are the closest real solutions to this problem.