15. find the x-intercept by factoring.
y=x^2-4x-32
here is what i have
(x+4)*(x-8) =0
x=-4
x=8.
16. show how the value of the discriminant supports your conclusion from question 15.
17. How does the axis of symmetry relate to the x-intercept.
Thank you for the help.
16. b^2 - 4 a c = 144 ... a perfect square
... the roots (intercepts) are unequal integers
17. the equation of the axis of symmetry is the average of the intercepts
... a.o.s. ... x = (8 - 4) / 2
15 correct
16
the discriminant is b^2 - 4ac
for yours we get 16-4(1)(-32) = 144
which is a positive number and a perfect square,
so we have to different rational x-intercepts.
Remember if the discriminant is > zero , you have two real roots
if the discriminant is = zero, you have one root
if the discriminant is < zero, you have no real roots, the parabola would not cross the x-axis
17.
If you change the parabola to the vertex form , you get
y = (x - 2)^2 - 36 , (I assume you know how to do that)
the axis of symmetry is x-2 = 0
or
x = 2
which is the midpoint value between 8 and -4
15. To find the x-intercepts of a quadratic equation by factoring, you need to set the equation equal to zero and then factor it.
In the given equation, y = x^2 - 4x - 32, set y equal to zero:
0 = x^2 - 4x - 32
To factor the quadratic equation, you need to find two numbers that multiply to give -32 and add up to -4 (the coefficient of x). In this case, the numbers are -8 and 4.
Rewrite the equation using the factored form:
0 = (x + 4)(x - 8)
Now, set each factor equal to zero and solve for x:
x + 4 = 0 --> x = -4
x - 8 = 0 --> x = 8
Therefore, the x-intercepts of the equation y = x^2 - 4x - 32 are x = -4 and x = 8.
16. The discriminant of a quadratic equation is given by the formula Δ = b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation ax^2 + bx + c = 0.
In this case, the quadratic equation is x^2 - 4x - 32 = 0. Comparing this with the general form ax^2 + bx + c = 0, we have a = 1, b = -4, and c = -32.
Now, calculate the discriminant:
Δ = (-4)^2 - 4(1)(-32)
= 16 + 128
= 144
The value of the discriminant is 144.
Since the discriminant is positive, we can conclude that there are two distinct real roots or x-intercepts. This supports our conclusion from question 15, where we found two different x-intercepts (-4 and 8).
17. The axis of symmetry of a quadratic equation is given by the formula x = -b/2a. It represents the vertical line that passes through the vertex of the parabola.
In this case, the quadratic equation is y = x^2 - 4x - 32. Comparing this with the form y = ax^2 + bx + c, we have a = 1, b = -4.
To find the axis of symmetry, substitute these values into the formula:
x = -(-4) / (2*1)
x = 4/2
x = 2
The axis of symmetry is x = 2.
The x-intercepts of the quadratic equation occur where y is equal to zero. Since the parabola is symmetric about the axis of symmetry, the x-intercepts will be equidistant from the axis. In other words, their x-values will have the same distance from the axis of symmetry, but in opposite directions.
In this case, the axis of symmetry is x = 2. The x-intercepts we found in question 15 are x = -4 and x = 8. These values are equidistant from the axis of symmetry, as the distance from 2 to -4 is the same as the distance from 2 to 8.