given (x+a)(x+b)=2x^2+20x+48, what is the value of a^2+b^2?
Please help. Thank you!
(x+a)(x+b)=2x^2+20x+48
if (x+a)(x+b) = 0 , sum of roots = -a - b, product of roots = ab
if 2x^2 + 20x + 48 = 0 , sum of roots = -20/2 = -10, product of roots = 48/2 = 24
-a - b = -10
a+b = 10
b = 10-b
ab = 24
a(10-b) = 24
10a - a^2 - 24 = 0
a^2 - 10a + 24 = 0
(a-6)(a-4) = 0
a = 6 or a = 4
if a=6, then b = 4
if a = 4, then b = 6
in either case a^2 + b^2 = 16 + 36 = 52
Using Reiny's fine introduction, we can proceed as follow:
(a+b)^2 = a^2 + 2ab + b^2 = a^2+b^2 + 2ab
(-10)^2 = a^2+b^2 + 2*24
100 = a^2+b^2 + 48
a^2+b^2 = 52
Thank you so much Mr. Reiny!
Thank you so much oobleck!
To find the value of a² + b², we need to first expand the given expression (x+a)(x+b).
(x+a)(x+b) = x(x+b) + a(x+b)
= x² + xb + ax + ab
= x² + (a+b)x + ab
Now, we can compare the expanded expression with the given quadratic equation 2x² + 20x + 48.
x² + (a+b)x + ab = 2x² + 20x + 48
By comparing the coefficients of the like terms, we can write the following equations:
1. Coefficient of x²: 1 = 2
2. Coefficient of x: a+b = 20
3. Constant term: ab = 48
From the first equation, we can determine the value of x.
2x² - x² = 20x - 2x²
x² = 20x
x = 20
Now, substituting the value of x = 20 into the second equation:
a+b = 20
Since we don't have any specific values for a and b, we cannot directly find their individual values from this equation. However, we can find their sum.
We need to find a² + b². To do this, we need to square the equation a+b = 20:
(a+b)² = 20²
(a+b)² = 400
Expanding (a+b)², we get:
(a+b)² = a² + 2ab + b²
Substituting the value of ab from the third equation, we have:
a² + 2(48) + b² = 400
a² + 96 + b² = 400
a² + b² = 400 - 96
a² + b² = 304
Therefore, the value of a² + b² is 304.