# Modelling a quadratic - Systems of linear equations

#### Ari

Hi, i cant figure out how to set up this question:
ax^2 + bx + c passes through (-1,5) and has a horizontal tangent at (1,7) find the coefficents a b and c

Thanks.

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#### dwsmith

MHF Hall of Honor
Hi, i cant figure out how to set up this question:
ax^2 + bx + c passes through (-1,5) and has a horizontal tangent at (1,7) find the coefficents a b and c

Thanks.
$$\displaystyle a(-1)^2-1b+c=5\rightarrow a-b+c=5$$

derivative

$$\displaystyle 2xa+b=0$$ when x=-1

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#### Ari

k im confused, how does that help me find the the values for a,b,c?

edit: and can you please explain how you got that? i feel very confused atm lol.

#### dwsmith

MHF Hall of Honor
Setup the coefficient matrix for the two equations and solve simultaneously.

#### Ari

Setup the coefficient matrix for the two equations and solve simultaneously.
i just started this course... which two equations are you talking about?

#### dwsmith

MHF Hall of Honor
i just started this course... which two equations are you talking about?
1: $$\displaystyle a-b+c=5$$
2: $$\displaystyle -2a+b=0$$

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#### Ari

1: $$\displaystyle a-b+c=5$$
2: $$\displaystyle 2a+b=0$$
oh okay, how did you get that second equations from the

$$\displaystyle a-b+c=5$$\

edit: nevermind i see it now. its from the originial one.

now how exactly do i solve them simultaneously? i can get the parametric solutions, but how do i get the number values?

#### dwsmith

MHF Hall of Honor
$$\displaystyle a(-1)^2-1b+c=5\rightarrow a-b+c=5$$

derivative

$$\displaystyle 2xa+b=0\rightarrow -2a+b=0$$ when x=-1
The 2nd equation came from the original equation.

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#### Ari

is the answer just going to be a,b,c in parametric form or will there be coefficent values, if so how do i get values for them?

#### dwsmith

MHF Hall of Honor
is the answer just going to be a,b,c in parametric form or will there be coefficent values, if so how do i get values for them?
Reduced row echelon form of the coefficient matrix.

$$\displaystyle \begin{bmatrix} 1 & -1 & 1 & 5\\ -2 & 1 & 0 & 0 \end{bmatrix}$$

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