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Sagot :
Certainly! Let's break the problem down step-by-step:
### Part (a): Explain why Sadie is wrong
Sadie suggests that [tex]\( p = -12 \)[/tex] and [tex]\( q = 10 \)[/tex].
Let's verify her solution with the given conditions.
#### Checking the Sum:
[tex]\[ p + q = -12 + 10 = -2 \][/tex]
This satisfies the condition [tex]\( p + q = -2 \)[/tex].
#### Checking the Product:
[tex]\[ p \cdot q = -12 \cdot 10 = -120 \][/tex]
This does not satisfy the condition [tex]\( pq = -8 \)[/tex].
Since [tex]\(-120 \neq -8\)[/tex], Sadie's proposed values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] do not satisfy both conditions. Therefore, Sadie is wrong.
### Part (b): Work out the correct values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]
We need to find values for [tex]\( p \)[/tex] and [tex]\( q \)[/tex] that satisfy the conditions:
[tex]\[ p + q = -2 \][/tex]
[tex]\[ pq = -8 \][/tex]
These conditions form a system of equations and can be interpreted as coefficients of a quadratic equation:
[tex]\[ t^2 - (p+q)t + pq = 0 \][/tex]
Substituting [tex]\( p+q = -2 \)[/tex] and [tex]\( pq = -8 \)[/tex]:
[tex]\[ t^2 + 2t - 8 = 0 \][/tex]
We solve this quadratic equation to find the roots, which will give us [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
#### Solving the Quadratic Equation:
The equation is:
[tex]\[ t^2 + 2t - 8 = 0 \][/tex]
Using the quadratic formula [tex]\( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -8 \)[/tex]:
[tex]\[ t = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \][/tex]
[tex]\[ t = \frac{-2 \pm \sqrt{4 + 32}}{2} \][/tex]
[tex]\[ t = \frac{-2 \pm \sqrt{36}}{2} \][/tex]
[tex]\[ t = \frac{-2 \pm 6}{2} \][/tex]
This yields two solutions:
[tex]\[ t = \frac{-2 + 6}{2} = 2 \][/tex]
[tex]\[ t = \frac{-2 - 6}{2} = -4 \][/tex]
Therefore, the values for [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are:
[tex]\[ p = 2 \][/tex]
[tex]\[ q = -4 \][/tex]
Since [tex]\( p > q \)[/tex], we confirm that [tex]\( p = 2 \)[/tex] and [tex]\( q = -4 \)[/tex] are indeed the correct values.
So, the correct values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are [tex]\( p = 2 \)[/tex] and [tex]\( q = -4 \)[/tex].
### Part (a): Explain why Sadie is wrong
Sadie suggests that [tex]\( p = -12 \)[/tex] and [tex]\( q = 10 \)[/tex].
Let's verify her solution with the given conditions.
#### Checking the Sum:
[tex]\[ p + q = -12 + 10 = -2 \][/tex]
This satisfies the condition [tex]\( p + q = -2 \)[/tex].
#### Checking the Product:
[tex]\[ p \cdot q = -12 \cdot 10 = -120 \][/tex]
This does not satisfy the condition [tex]\( pq = -8 \)[/tex].
Since [tex]\(-120 \neq -8\)[/tex], Sadie's proposed values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] do not satisfy both conditions. Therefore, Sadie is wrong.
### Part (b): Work out the correct values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex]
We need to find values for [tex]\( p \)[/tex] and [tex]\( q \)[/tex] that satisfy the conditions:
[tex]\[ p + q = -2 \][/tex]
[tex]\[ pq = -8 \][/tex]
These conditions form a system of equations and can be interpreted as coefficients of a quadratic equation:
[tex]\[ t^2 - (p+q)t + pq = 0 \][/tex]
Substituting [tex]\( p+q = -2 \)[/tex] and [tex]\( pq = -8 \)[/tex]:
[tex]\[ t^2 + 2t - 8 = 0 \][/tex]
We solve this quadratic equation to find the roots, which will give us [tex]\( p \)[/tex] and [tex]\( q \)[/tex].
#### Solving the Quadratic Equation:
The equation is:
[tex]\[ t^2 + 2t - 8 = 0 \][/tex]
Using the quadratic formula [tex]\( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -8 \)[/tex]:
[tex]\[ t = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-8)}}{2 \cdot 1} \][/tex]
[tex]\[ t = \frac{-2 \pm \sqrt{4 + 32}}{2} \][/tex]
[tex]\[ t = \frac{-2 \pm \sqrt{36}}{2} \][/tex]
[tex]\[ t = \frac{-2 \pm 6}{2} \][/tex]
This yields two solutions:
[tex]\[ t = \frac{-2 + 6}{2} = 2 \][/tex]
[tex]\[ t = \frac{-2 - 6}{2} = -4 \][/tex]
Therefore, the values for [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are:
[tex]\[ p = 2 \][/tex]
[tex]\[ q = -4 \][/tex]
Since [tex]\( p > q \)[/tex], we confirm that [tex]\( p = 2 \)[/tex] and [tex]\( q = -4 \)[/tex] are indeed the correct values.
So, the correct values of [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are [tex]\( p = 2 \)[/tex] and [tex]\( q = -4 \)[/tex].
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