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Could i have super quick help before i'm out of time?

Could i have super quick help before i'm out of time?-example-1
User Salaam
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1 Answer

12 votes
12 votes

Answer:

Part (a)

Given quadratic:
y=x^2+2x-8

Factored form

To factor, find two numbers that multiply to -8 and sum to 2: 4 and -2

Rewrite the middle term of the quadratic as the sum of these number:


\implies y=x^2+4x-2x-8

Factorize the first two terms and the last two terms separately:


\implies y=x(x+4)-2(x+4)

Factor out the common term
(x+4):


\implies y=(x-2)(x+4)

Zeros

The zeros of the quadratic polynomial are the x-coordinates of the points where the graph intersects the x-axis, i.e. when y = 0


\implies y=0


\implies (x-2)(x+4)=0


\implies (x-2)=0\implies x=2


\implies (x+4)=0\implies x=-4

Therefore, the zeros are 2 and -4

Vertex

The x-coordinate of the vertex is the midpoint of the zeros.


\textsf{midpoint}=(-4+2)/(2)=-1

To find the y-coordinate of the vertex, substitute the found value of x into the given equation:


y=(-1)^2+2(-1)-8=-9

Therefore, the vertex is (1, -9)

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Part (b)

Given quadratic:
y=-x^2-9x-14

Factored form

To factor, first factor out -1:


y=-(x^2+9x+14)

Now find two numbers that multiply to 14 and sum to 9: 7 and 2

Rewrite the middle term of the quadratic as the sum of these number:


y=-(x^2+2x+7x+14)

Factorize the first two terms and the last two terms separately:


y=-(x(x+2)+7(x+2))

Factor out the common term
(x+2):


y=-(x+7)(x+2)

Zeros

The zeros of the quadratic polynomial are the x-coordinates of the points where the graph intersects the x-axis, i.e. when y = 0


\implies y=0


\implies -(x+7)(x+2)=0


\implies -(x+7)=0 \implies x=-7


\implies (x+2)=0 \implies x=-2

Therefore, the zeros are -7 and -2

Vertex

The x-coordinate of the vertex is the midpoint of the zeros.


\textsf{midpoint}=(-7+(-2))/(2)=-4.5

To find the y-coordinate of the vertex, substitute the found value of x into the given equation:


y=-(-4.5)^2-9(-4.5)-14=6.25

Therefore, the vertex is (-4.5, 6.25)

Could i have super quick help before i'm out of time?-example-1
Could i have super quick help before i'm out of time?-example-2
User Jordi Corominas
by
2.3k points