To find two numbers that multiply to $-6$ and add to $4$, let the numbers be $x$ and $y$. These two numbers must satisfy the equations:

$x \cdot y = -6$ $x + y = 4$

Steps to Solve:

1. From $x + y = 4$, we can express $y$ in terms of $x$: $y = 4 - x$

2. Substitute $y = 4 - x$ into $x \cdot y = -6$: $x \cdot (4 - x) = -6$

3. Expand and simplify: $4x - x^2 = -6$ $x^2 - 4x - 6 = 0$

4. Solve the quadratic equation $x^2 - 4x - 6 = 0$ using the quadratic formula: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ Here, $a = 1$, $b = -4$, and $c = -6$: $x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(-6)}}{2(1)}$ $x = \frac{4 \pm \sqrt{16 + 24}}{2}$ $x = \frac{4 \pm \sqrt{40}}{2}$ $x = \frac{4 \pm 2\sqrt{10}}{2}$ $x = 2 \pm \sqrt{10}$

Solutions:

• The two numbers are: $x = 2 + \sqrt{10}, \quad y = 2 - \sqrt{10}$ OR $x = 2 - \sqrt{10}, \quad y = 2 + \sqrt{10}$

These numbers multiply to $-6$ and add to $4$.

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Do you want further details or help visualizing this solution?

Here are 5 related questions to explore:

1. How can we graphically represent the solution of $x^2 - 4x - 6 = 0$?
2. How is the quadratic formula derived, and when is it used?
3. Can we estimate $\sqrt{10}$ to approximate the answers numerically?
4. How can we check if the two numbers truly satisfy both conditions?
5. What is the significance of the discriminant in the quadratic formula?

Tip: Always check your solutions by substituting them back into the original equations!