Question

question 1
convert the expression ((x + 2.5)^2 - 56.25) into standard and factored form.
standard form:
factored form:
question 2
convert the expression ((x + 9)(x - 7)) into vertex and standard form.
vertex form:
standard form:
question 3
convert the expression ((x - 2)(x - 6)) into vertex and standard form.
vertex form:
standard form:

Explanation:

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Question 1

Step1: Expand the squared term

$(x+2.5)^2 = x^2 + 5x + 6.25$

Step2: Subtract 56.25 to get standard form

$x^2 + 5x + 6.25 - 56.25 = x^2 + 5x - 50$

Step3: Rewrite as difference of squares

$(x+2.5)^2 - 56.25 = (x+2.5)^2 - (7.5)^2$

Step4: Apply difference of squares rule

$(x+2.5-7.5)(x+2.5+7.5) = (x-5)(x+10)$

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Question 2

Step1: Expand to get standard form

$(x+9)(x-7) = x^2 -7x +9x -63 = x^2 +2x -63$

Step2: Complete the square for vertex form

$x^2 +2x = (x+1)^2 -1$

Step3: Substitute back to find vertex form

$(x+1)^2 -1 -63 = (x+1)^2 -64$

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Question 3

Step1: Expand to get standard form

$(x-2)(x-6) = x^2 -6x -2x +12 = x^2 -8x +12$

Step2: Complete the square for vertex form

$x^2 -8x = (x-4)^2 -16$

Step3: Substitute back to find vertex form

$(x-4)^2 -16 +12 = (x-4)^2 -4$

Answer:

1. standard form: $x^2 + 5x - 50$

factored form: $(x-5)(x+10)$

1. vertex form: $(x+1)^2 - 64$

standard form: $x^2 + 2x - 63$

1. vertex form: $(x-4)^2 - 4$

standard form: $x^2 - 8x + 12$