Mathematics: Determine the equation of the quadratic with zeros x = 1 and x = 5 passing through the point (3,-12). Please help

49.92Step-by-step explanation:...

Determine the equation of the quadratic with zeros

blesskids600

29.01.2022

$y=3x^{2} -18x+15$ is the equation the quadratic as per the given conditions.

Step-by-step explanation:

Let the equation of the quadratic be:

$y=ax^{2} +bx+c$

Given that it has zeroes at x = 1 and x = 5 i.e. y = 0 at both the values of x.

Also passes through point (3, -12). i.e. when x = 3, y = -12

Putting x = 1, y = 0:

$\Rightarrow a\times 1^{2} +b\times 1+c=0\\\Rightarrow a+b+c=0 ....... (1)$

Putting x = 5, y = 0:

$\Rightarrow a\times 5^{2} +b\times 5+c=0\\\Rightarrow 25a+5b+c=0 ....... (2)$

Putting x = 3, y = -12:

$\Rightarrow a\times 3^{2} +b\times 3+c=-12\\\Rightarrow 9a+3b+c=-12 ....... (3)$

Equation (2) - Equation (1):

$24a+4b=0\\\Rightarrow 6a +b=0 ...... (4)$

Equation (2) - Equation (3):

$16a+2b=12\\\Rightarrow 8a +b=6 ...... (5)$

Equation (5) - Equation (4):

$2a=6\\\Rightarrow a =3$

Putting value of a in equation (4):

$6\times 3+b=0\\\Rightarrow b = -18$

Putting a and b in equation (1):

$3-18+c=0\\\Rightarrow c= 15$

So, the quadratic equation is:

$y=3x^{2} -18x+15$

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