Question

Which two numbers when multiplied gives negative 3 but add to give positive 8

Answer 1

To find two numbers that multiply to give -3 and add to give +8, we can use the equations:

1. \( x \cdot y = -3 \)
2. \( x + y = 8 \)

Let's express \( y \) in terms of \( x \) using the second equation:

\[ y = 8 - x \]

Now, substitute this expression for \( y \) into the first equation:

\[ x(8 - x) = -3 \]

Expanding this gives us:

\[ 8x - x^2 = -3 \]

Rearranging the equation leads to:

\[ x^2 - 8x - 3 = 0 \]

Now we can use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 1, b = -8, c = -3 \):

1. Calculate the discriminant:

\[ b^2 - 4ac = (-8)^2 - 4(1)(-3) = 64 + 12 = 76 \]

2. Use the quadratic formula:

\[ x = \frac{8 \pm \sqrt{76}}{2} \]

Since \( \sqrt{76} = 2\sqrt{19} \), we can simplify:

\[ x = \frac{8 \pm 2\sqrt{19}}{2} = 4 \pm \sqrt{19} \]

Thus, the two numbers are:

1. \( x = 4 + \sqrt{19} \)
2. \( y = 4 - \sqrt{19} \)

Verifying:

- Their product:

\[
(4 + \sqrt{19})(4 - \sqrt{19}) = 16 - 19 = -3
\]

- Their sum:

\[
(4 + \sqrt{19}) + (4 - \sqrt{19}) = 8
\]

So, the two numbers are \( 4 + \sqrt{19} \) and \( 4 - \sqrt{19} \).