The vertex form of a quadratic equation is a (x - h)^(2) + ka(x−h)2+k.
We have: y = (6 x + 3)(x - 5)y=(6x+3)(x−5)
To express this equation in its vertex form, we must "complete the square".
First, let's expand the parentheses:
Rightarrow y = 6 x^(2) - 30 x + 3 x - 15⇒y=6x2−30x+3x−15
Rightarrow y = 6 x^(2) - 27 x - 15⇒y=6x2−27x−15
Then, let's factor 66 out of the equation:
Rightarrow y = 6 (x^(2) - frac(27)(6) x - frac(15)(6))⇒y=6(x2−276x−156)
Rightarrow y = 6 (x^(2) - frac(9)(2) x - frac(5)(2))⇒y=6(x2−92x−52)
Now, let's add and subtract the square of half of the xx term within the parentheses:
Rightarrow y = 6 (x^(2) - frac(9)(2) x + (frac(9)(4))^(2) - frac(5)(2) - (frac(9)(4))^(2))⇒y=6(x2−92x+(94)2−52−(94)2)
Rightarrow y = 6 ((x - frac(9)(4))^(2) - frac(5)(2) - frac(81)(16))⇒y=6((x−94)2−52−8116)
Rightarrow y = 6 ((x - frac(9)(4))^(2) - frac(121)(16))⇒y=6((x−94)2−12116)
Finally, let's distribute 66 throughout the parentheses:
therefore = 6 (x - frac(9)(4))^(2) - frac(363)(8)
