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.
solve the system of equations y=2x^2-3 y=3x-1
7 answers
how many real numbers solutions are there to the equation 0=-3x^2+x-4
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.
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.
solve the equation by completing the square if necessary round to the nearest hundredth x^2-18x=19
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.
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.
solve x^2-81=0
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.
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.
2 answers