Question

solve the system of equations y=2x^2-3 y=3x-1

Answer 1

[AI-generated answer — Bot GPT 3.5]

To solve the system of equations:

y = 2x^2 - 3
y = 3x - 1

We can set the two equations equal to each other:

2x^2 - 3 = 3x - 1

Rearranging the equation:

2x^2 - 3x + 2 = 0

To solve this quadratic equation, we can use factoring, completing the square, or the quadratic formula. In this case, let's use the quadratic formula:

x = (-b ± √(b^2 - 4ac)) / (2a)

For our equation:

a = 2, b = -3, and c = 2

Substituting these values into the quadratic formula:

x = (-(-3) ± √((-3)^2 - 4(2)(2))) / (2(2))
x = (3 ± √(9 - 16)) / 4
x = (3 ± √(-7)) / 4

Since we have a negative value under the square root, the equation has no real solutions. Therefore, the system of equations has no real solutions.

Answer 2

how many real numbers solutions are there to the equation 0=-3x^2+x-4

Answer 3

[AI-generated answer — Bot GPT 3.5]

To determine the number of real number solutions to the equation:

0 = -3x^2 + x - 4

We can use the discriminant (b^2 - 4ac) of the quadratic equation. In this case, the equation is in the form ax^2 + bx + c = 0.

Comparing our equation to this form, we have:

a = -3
b = 1
c = -4

The discriminant is given by:

D = b^2 - 4ac

Substituting the values:

D = (1)^2 - 4(-3)(-4)
D = 1 - 48
D = - 47

Since the discriminant is negative (D < 0), there are no real number solutions to the equation.

Answer 4

solve the equation by completing the square if necessary round to the nearest hundredth x^2-18x=19

Answer 5

[AI-generated answer — Bot GPT 3.5]

To solve the equation by completing the square:

x^2 - 18x = 19

First, we need to rewrite the equation so that the x terms are grouped together. We can do this by adding (18/2)^2 = 81 to both sides of the equation:

x^2 - 18x + 81 = 19 + 81
x^2 - 18x + 81 = 100

Now, we can express the left side of the equation as a perfect square:

(x - 9)^2 = 100

Taking the square root of both sides:

√((x - 9)^2) = ±√(100)

Simplifying:

x - 9 = ±10

To solve for x, we can separate the equation into two cases:

Case 1:
x - 9 = 10
x = 10 + 9
x = 19

Case 2:
x - 9 = -10
x = -10 + 9
x = -1

Therefore, the equation has two solutions: x = 19 and x = -1.

Answer 6

solve x^2-81=0

Answer 7

[AI-generated answer — Bot GPT 3.5]

To solve the equation x^2 - 81 = 0:

We can rewrite the equation as (x - 9)(x + 9) = 0.

From here, we can set each factor equal to zero and solve for x:

x - 9 = 0 or x + 9 = 0

For x - 9 = 0, adding 9 to both sides gives us:

x = 9

For x + 9 = 0, subtracting 9 from both sides gives us:

x = -9

Therefore, the equation has two solutions: x = 9 and x = -9.