Match each quadratic function given in factored form with its equivalent standard form listed.
$f(x)=x^2-x-12\left|\sharp \quad f(x)=x^2-11 x-12\right| \sharp \quad f(x)=x^2+x-12 \mid \quad \exists$
$f(x)=x^2-4 x-12$
$f(x)=(x-4)(x+3)$
$f(x)=(x-3)(x+4)$
$f(x)=(x-12)(x+1)$
$f(x)=(x+2)(x-6)$
Answer (1)
Expand each quadratic function from its factored form to its standard form.
( x − 4 ) ( x + 3 ) expands to x 2 − x − 12 .
( x − 3 ) ( x + 4 ) expands to x 2 + x − 12 .
( x − 12 ) ( x + 1 ) expands to x 2 − 11 x − 12 .
( x + 2 ) ( x − 6 ) expands to x 2 − 4 x − 12 .
Match the expanded forms to the given standard forms: ( x − 4 ) ( x + 3 ) = x 2 − x − 12 ( x − 3 ) ( x + 4 ) = x 2 + x − 12 ( x − 12 ) ( x + 1 ) = x 2 − 11 x − 12 ( x + 2 ) ( x − 6 ) = x 2 − 4 x − 12
Explanation
Problem Analysis We are given four quadratic functions in factored form and four in standard form. Our goal is to match each factored form with its equivalent standard form. We will do this by expanding each factored form and comparing it to the standard forms.
Expanding (x-4)(x+3) Let's expand the first factored form: f ( x ) = ( x − 4 ) ( x + 3 ) . Using the distributive property (also known as FOIL), we have: ( x − 4 ) ( x + 3 ) = x ( x ) + x ( 3 ) − 4 ( x ) − 4 ( 3 ) = x 2 + 3 x − 4 x − 12 = x 2 − x − 12
Expanding (x-3)(x+4) Now, let's expand the second factored form: f ( x ) = ( x − 3 ) ( x + 4 ) .
( x − 3 ) ( x + 4 ) = x ( x ) + x ( 4 ) − 3 ( x ) − 3 ( 4 ) = x 2 + 4 x − 3 x − 12 = x 2 + x − 12
Expanding (x-12)(x+1) Next, we expand the third factored form: f ( x ) = ( x − 12 ) ( x + 1 ) .
( x − 12 ) ( x + 1 ) = x ( x ) + x ( 1 ) − 12 ( x ) − 12 ( 1 ) = x 2 + x − 12 x − 12 = x 2 − 11 x − 12
Expanding (x+2)(x-6) Finally, let's expand the fourth factored form: f ( x ) = ( x + 2 ) ( x − 6 ) .
( x + 2 ) ( x − 6 ) = x ( x ) + x ( − 6 ) + 2 ( x ) + 2 ( − 6 ) = x 2 − 6 x + 2 x − 12 = x 2 − 4 x − 12
Matching the Forms Now we match the expanded forms with the given standard forms:
( x − 4 ) ( x + 3 ) = x 2 − x − 12
( x − 3 ) ( x + 4 ) = x 2 + x − 12
( x − 12 ) ( x + 1 ) = x 2 − 11 x − 12
( x + 2 ) ( x − 6 ) = x 2 − 4 x − 12
So the matches are:
f ( x ) = ( x − 4 ) ( x + 3 ) matches f ( x ) = x 2 − x − 12
f ( x ) = ( x − 3 ) ( x + 4 ) matches f ( x ) = x 2 + x − 12
f ( x ) = ( x − 12 ) ( x + 1 ) matches f ( x ) = x 2 − 11 x − 12
f ( x ) = ( x + 2 ) ( x − 6 ) matches f ( x ) = x 2 − 4 x − 12
Examples
Understanding how to convert between factored and standard forms of quadratic equations is useful in many real-world applications. For example, engineers use quadratic equations to model the trajectory of projectiles, such as the path of a ball thrown through the air. The factored form can help determine the points where the projectile hits the ground, while the standard form can help determine the maximum height of the projectile. By understanding these forms, engineers can design more efficient and accurate systems.
