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]
