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If f(x)=ax^2+b and 3f(x)+2f(-x)=5x^2-10 for all x is f(x)=7?
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20 Feb 2024, 03:50
20 Feb 2024, 03:50
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If \(f(x)=ax^2+b\) and \(3f(x)+2f(-x)=5x^2-10\) for all x, for what value of x is \(f(x)=7\)?
Indicate all that apply
A. -9
B. -3
C. 0
D. 3
E. 4
Indicate all that apply
A. -9
B. -3
C. 0
D. 3
E. 4
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Free Materials for the GRE General Exam - Where to get it!!
GRE Geometry Formulas
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If f(x)=ax^2+b and 3f(x)+2f(-x)=5x^2-10 for all x is f(x)=7?
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20 Feb 2024, 09:38
20 Feb 2024, 09:38
1
\(3f(x) = 3(ax^2+ b) = 3ax^2+3b\)
\(2f(-x) = 2(a(-x)^2+b) = 2ax^2 +2b\)
Combine to get \(3f(x)+2f(-x)=5ax^2+5b\) and set it equal to the given \(5x^2-10\)
\(5ax^2+5b=5x^2-10\)
a = 1, b = -2
Plug back into the original f(x) and set it to equal 7 to find the solution for x
\((1)x^2+(-2) = 7\)
\(x^2 = 9\)
x = -3 and 3
\(2f(-x) = 2(a(-x)^2+b) = 2ax^2 +2b\)
Combine to get \(3f(x)+2f(-x)=5ax^2+5b\) and set it equal to the given \(5x^2-10\)
\(5ax^2+5b=5x^2-10\)
a = 1, b = -2
Plug back into the original f(x) and set it to equal 7 to find the solution for x
\((1)x^2+(-2) = 7\)
\(x^2 = 9\)
x = -3 and 3
Re: If f(x)=ax^2+b and 3f(x)+2f(-x)=5x^2-10 for all x is f(x)=7?
[#permalink]
24 Sep 2024, 14:58
24 Sep 2024, 14:58
1
Substitute for x and -x
3(ax^2 + b) + 2(ax^2 + b) = 5x^2 - 10
3ax^2 + 3b + 2ax^2 + 2b = 5x^2 - 10
5ax^2 + 5b = 5( x^2 - 2)
5( ax^2 + b ) = 5(x^2 - 2)
ax^2 + b = x^2 - 2
Since f(x) = 7
x^2 - 2 = 7
x^2 = 9
x = 3 or -3
Answer B and D
3(ax^2 + b) + 2(ax^2 + b) = 5x^2 - 10
3ax^2 + 3b + 2ax^2 + 2b = 5x^2 - 10
5ax^2 + 5b = 5( x^2 - 2)
5( ax^2 + b ) = 5(x^2 - 2)
ax^2 + b = x^2 - 2
Since f(x) = 7
x^2 - 2 = 7
x^2 = 9
x = 3 or -3
Answer B and D
