Question

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Answer 1

Answer:

The given quadratic has two solutions.

Step-by-step explanation:

The discriminant of a quadratic equation is [(b^2) - (4 * a * c)]

We're given the quadratic x^2 - 8x - 84 = 0

We can find a, b, and c by analyzing the quadratic.

So,

a = 1 because x^2 has no coeffiecient

b = -8 ( coefficient of 8 * the - )

c = -84 the raw y-intercept value of this function

Now we can plug these numbers into the discriminant expression:

(-8)^2 - ( 4 * 1 * -84 ) =

64 - ( - 336 ) =

64 + 336 =

400

Now that we know the value of the discriminant, we have to anazlyze it.

If the discriminant of a quadratic is positive, the quadratic has 2 solutions.

If the discriminant is negative, the quadratic has no solutions.

If the discriminant is equal to zero, the quadratic only has 1 solution.

In conclusion, because the discriminant = 400, and 400 > 0, the given quadratic equation has two solutions.

Answer 2

Answer: x=(-6,14)

Steps:
Factor the expression.

Apply the AC Method
Look at the coefficients, A=1,B=-8,C=-84

Find factors of AC.
{1,2,3,4,6,7,12,14,21,28,42,84}

Find pairs of the above factors, which, when multiplied equal 84.
{1*84,2*42,3*28,4*21,6*14,7*12}

Which pairs of these factors have a difference of B = -8?
6-14=-8

Split the middle term to use above pair.
x^2+(6×-14x)-84

Factor x out of the first pair.
x*(x+6)+(-14x-84)

Factor-14 out of the second pair.
x*(x+6)-14*(x+6)

Group the common factor.
(x-14)*(x+6)

Set 1 st factor x-14 to 0 to solve.
x-14=0

Add 14 to both sides
x-14+14=0+14
Simplify
x=14

Set 2nd factor x+6 to 0 to solve.
x+6=0

Subtract 6 from both sides
x*6-6=0-6
Simplify
x=-6

Solution:
x=(-6,14)