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Sagot :
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: 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)
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)
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