What is the minimum value of the expression 2x^2+3y^2+8x-24y+62 for real x and y?
To find the minimum value of the expression 2x^2 + 3y^2 + 8x - 24y + 62 for real values of x and y, we can use a mathematical technique called completing the square.
First, let's focus on the x terms. We can rewrite the expression as follows:
2x^2 + 8x + (3y^2 - 24y) + 62
To complete the square for the x terms, we need to add and subtract a constant. In this case, the constant we want to add is (8/2)^2 = 16. However, since we're adding this constant to the expression, we also need to subtract it to maintain the original value.
Let's rewrite the expression, completing the square for the x terms:
2x^2 + 8x + 16 - 16 + (3y^2 - 24y) + 62
(2x^2 + 8x + 16) - 16 + (3y^2 - 24y) + 62
Now, let's do the same for the y terms. We need to add and subtract a constant to complete the square. This time, the constant we want to add is (24/2)^2 = 144. Again, we'll subtract this constant as well.
Let's rewrite the expression, completing the square for both x and y terms:
(2x^2 + 8x + 16) - 16 + (3y^2 - 24y + 144) - 144 + 62
(2x^2 + 8x + 16) + (3y^2 - 24y + 144) - 98
Now, we can group the x terms and y terms separately. Simplify each group separately:
2(x^2 + 4x + 8) + 3(y^2 - 8y + 48) - 98
To find the minimum value, we want to make each group of terms as small as possible. The minimum value for each parentheses can be found by using the formula (a+b)^2 = a^2 + 2ab + b^2.
For the x terms, the minimum value is when x^2 + 4x + 8 is equal to zero. To make it equal to zero, we need to find the x-coordinate of the vertex of the quadratic. The x-coordinate of the vertex can be found using the formula x = -b/2a.
In this case, a = 1, and b = 4. So, the x-coordinate of the vertex is x = -4/(2*1) = -2.
Now, let's substitute this value back into the expression to find the minimum value for the x terms:
2(-2^2 + 4(-2) + 8) = 2(-4 - 8 + 8) = 2(-4) = -8
For the y terms, the minimum value is when y^2 - 8y + 48 is equal to zero. Similarly, we can find the y-coordinate of the vertex by using the formula y = -b/2a.
In this case, a = 1, and b = -8. So, the y-coordinate of the vertex is y = -(-8)/(2*1) = 4.
Now, let's substitute this value back into the expression to find the minimum value for the y terms:
3(4^2 - 8(4) +48) = 3(16 - 32 +48) = 3(32) = 96
Finally, let's substitute the minimum values we found for the x and y terms back into the expression:
-8 + 96 - 98 = -8 + 96 - 98 = -10
Therefore, the minimum value of the expression 2x^2 + 3y^2 + 8x - 24y + 62 for real x and y is -10.