Value of a a and b b

Algebra Level 2

Quadratic function f ( x ) = a x 2 + b x + c f(x) = ax^{2} + bx + c is such that f ( x + 2 ) − f ( x ) = 4 x + 2 f(x + 2) - f(x) = 4x + 2 . Find the value of a a and b b .

a = − 2 , b = − 1 a = -2, b = -1 a = 1 , b = − 1 a = 1, b = -1 a = 2 , b = 1 a = 2, b = 1 a = 1 , b = 2 a = 1, b = 2 a = − 1 , b = 1 a = -1, b = 1 a = − 1 , b = − 2 a = -1, b = -2

The Strategy Gamer by The Strategy Gamer , 17, Philippines

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

2 solutions

Chew-Seong Cheong
Sep 28, 2017

f ( x ) = a x 2 + b x + c f ( x + 2 ) = a ( x 2 + 4 x + 4 ) + b ( x + 2 ) + c f ( x + 2 ) − f ( x ) = 4 a x + 4 a + 2 b Given that f ( x + 2 ) − f ( x ) = 4 x + 2 ⟹ 4 x + 2 = 4 a x + 4 a + 2 b \begin{aligned} f(x) & = ax^2 + bx + c \\ f(x+2) & = a(x^2+4x+4) + b(x+2) + c \\ \color{#3D99F6} f(x+2) - f(x) & = 4ax + 4a + 2b & \small \color{#3D99F6} \text{Given that }f(x+2)-f(x) = 4x+2 \\ \implies \color{#3D99F6} 4x + 2 & = 4ax + 4a + 2b \end{aligned}

Equating the coefficients ⟹ { 4 x = 4 a x ⟹ a = 1 2 = 4 a + 2 b = 4 + 2 b ⟹ b = − 1 \implies \begin{cases} 4x = 4ax & \implies a = 1 \\ 2 = 4a+2b = 4+2b & \implies b = -1 \end{cases}

The answer: a = 1 , b = − 1 \boxed{a=1, b=-1}

3 Helpful 0 Interesting 0 Brilliant 0 Confused
Patrick Zauner
Sep 28, 2017

If you plug in x + 2 x + 2 for x x and subtract f ( x ) f(x) you get 4 a x + 4 a + 2 b 4ax+4a+2b . Insert 1 for a a so you get the 4 x 4x , now you only have to plug in − 1 -1 for b b to get the + 2 +2 .

1 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...