Question

solve 2x^2 -12x -14 =0 by graphing the corresponding function and determing the zeros

Answer 1

To graph the corresponding function and determine the zeros, we first need to rewrite the equation in the form of "y = 0".

2x^2 - 12x - 14 = 0

Let's label this equation as f(x):

f(x) = 2x^2 - 12x - 14

To graph this function, we can plot points on a graph using different x-values and calculate the corresponding y-values. However, we can also use the quadratic formula to determine the zeros (x-intercepts) directly.

The quadratic formula states that for the equation ax^2 + bx + c = 0, the zeros are given by:

x = (-b ± √(b^2 - 4ac)) / (2a)

In our equation, a = 2, b = -12, and c = -14. Substituting these values into the quadratic formula:

x = (-(-12) ± √((-12)^2 - 4*2*(-14))) / (2*2)
= (12 ± √(144 + 112)) / 4
= (12 ± √256) / 4
= (12 ± 16) / 4

We have two solutions:

x1 = (12 + 16) / 4 = 28 / 4 = 7
x2 = (12 - 16) / 4 = -4 / 4 = -1

Thus, the zeros of the equation 2x^2 - 12x - 14 = 0 are x = 7 and x = -1.

To graph this function, we can plot these zeros on a coordinate plane:

(7,0) and (-1,0)

Then, to sketch the curve of the parabola, we can find additional points by substituting different x-values into the function and calculating the corresponding y-values.

For example, substituting x = 0:
f(0) = 2(0)^2 - 12(0) - 14
= 0 + 0 - 14
= -14

So, we have the point (0,-14).

Substituting x = 3:
f(3) = 2(3)^2 - 12(3) - 14
= 2(9) - 36 - 14
= 18 - 36 - 14
= -32

So, we have the point (3,-32).

Using the zeros and these additional points, we can sketch the graph of the function.