Answer :

Answer:

Step-by-step explanation:

Let's call the two numbers x and y.

Given:

1. The difference between the two numbers is 6: [tex]\(x - y = 6\)[/tex]

2. The product of the two numbers is 15: [tex]\(xy = 15\)[/tex]

We can solve this system of equations to find the values of x and y.

From equation (1), we can express x in terms of y:

[tex]\[x = y + 6\][/tex]

Substitute this expression for x into equation (2):

[tex]\[(y + 6)y = 15\][/tex]

Expand and rearrange:

[tex]\[y^2 + 6y = 15\]\[y^2 + 6y - 15 = 0\][/tex]

Now, we have a quadratic equation in terms of y. We can solve this equation using the quadratic formula:

[tex]\[y = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{{2a}}\][/tex]

Where:

[tex]a = 1 (coefficient of \(y^2\)) \\b = 6 (coefficient of y)\\c = -15 (constant term)[/tex]

Plugging in the values, we get:

[tex]\[y = \frac{{-6 \pm \sqrt{{6^2 - 4(1)(-15)}}}}{{2(1)}}\]\[y = \frac{{-6 \pm \sqrt{{36 + 60}}}}{{2}}\]\[y = \frac{{-6 \pm \sqrt{{96}}}}{{2}}\]\[y = \frac{{-6 \pm 4\sqrt{{6}}}}{{2}}\][/tex]

Now, we have two possible values for y:

[tex]\[y_1 = \frac{{-6 + 4\sqrt{{6}}}}{{2}}\]\[y_2 = \frac{{-6 - 4\sqrt{{6}}}}{{2}}\][/tex]

We can then use these values of y to find the corresponding values of x using the equation [tex]\(x = y + 6\).[/tex]

Let's calculate the values of x and y:

[tex]For \(y_1\):[/tex]

[tex]\[y_1 = \frac{{-6 + 4\sqrt{{6}}}}{2} \approx 2.12\]\[x_1 = y_1 + 6 \approx 8.12\][/tex]

[tex]For \(y_2\):[/tex]

[tex]\[y_2 = \frac{{-6 - 4\sqrt{{6}}}}{2} \approx -8.12\]\[x_2 = y_2 + 6 \approx -2.12\][/tex][tex]\[y_2 = \frac{{-6 - 4\sqrt{{6}}}}{2} \approx -8.12\]\[x_2 = y_2 + 6 \approx -2.12\][/tex]

So, the two numbers are approximately [tex]\(8.12\) and \(-2.12\), or \(-2.12\) and \(8.12\).[/tex]