What is the value of the discriminant, b2 − 4ac, for the quadratic equation 0 = −2x2 − 3x + 8, and what does it mean about the number of real solutions the equation has?
What are the solutions to the quadratic equation 4(x + 2)2 = 36
Mischa wrote the quadratic equation 0 = –x2 + 4x – 7 in standard form. What is the value of c in her equation?
To find the value of the discriminant, b^2 - 4ac, for a quadratic equation of the form ax^2 + bx + c = 0, we need to identify the values of a, b, and c.
For the quadratic equation 0 = -2x^2 - 3x + 8, we have a = -2, b = -3, and c = 8.
Now, plug these values into the discriminant formula:
b^2 - 4ac = (-3)^2 - 4(-2)(8)
Simplifying:
9 - (-64) = 9 + 64 = 73
The value of the discriminant is 73.
Now, let's interpret what this value tells us about the number of real solutions the equation has.
If the value of the discriminant is positive (in this case, 73), then the quadratic equation has two distinct real solutions.
If the value of the discriminant is zero, then the quadratic equation has one real solution (called a "double root").
If the value of the discriminant is negative, then the quadratic equation has no real solutions and only complex conjugate solutions.
Therefore, for the equation 0 = -2x^2 - 3x + 8, since the discriminant is positive (73), it means the equation has two distinct real solutions.
Moving on to the next question:
To find the solutions to the quadratic equation 4(x + 2)^2 = 36, we can follow these steps:
1. Begin by isolating the squared term by dividing both sides by 4:
(x + 2)^2 = 36 / 4
(x + 2)^2 = 9
2. Take the square root of both sides to eliminate the square:
x + 2 = ±√9
x + 2 = ±3
3. Solve for x by subtracting 2 from both sides:
x = -2 + 3 or x = -2 - 3
x = 1 or x = -5
The solutions to the quadratic equation 4(x + 2)^2 = 36 are x = 1 and x = -5.
Finally, let's determine the value of c in the quadratic equation 0 = -x^2 + 4x - 7, written in standard form.
In standard form, a quadratic equation is of the form ax^2 + bx + c = 0.
Comparing this to the given equation, we can see that a = -1, b = 4, and c is the coefficient of the constant term, which is -7.
Therefore, in Mischa's equation 0 = -x^2 + 4x - 7, the value of c is -7.
To find the value of the discriminant for the quadratic equation 0 = -2x^2 - 3x + 8, we can use the formula b^2 - 4ac.
Given that the coefficients are: a = -2, b = -3, and c = 8.
Plugging the values into the discriminant formula:
Discriminant = (-3)^2 - 4(-2)(8)
Discriminant = 9 - (-64)
Discriminant = 9 + 64
Discriminant = 73
The value of the discriminant is 73.
Now, let's analyze what this value means about the number of real solutions the equation has.
If the discriminant is positive (in this case, 73), then the quadratic equation has two distinct real solutions.
If the discriminant is zero, the quadratic equation has one real solution (a double root or repeated solution).
If the discriminant is negative, the quadratic equation has no real solutions, only complex solutions.
Therefore, for the given equation 0 = -2x^2 - 3x + 8, since the discriminant is positive (73 > 0), it means that the equation has two distinct real solutions.
the usual way to type a power is to use the ^ to show exponents,
e.g.
x^2 is "x squared"
0 = −2x^2 − 3x + 8 in standard form is
2x^2 + 3x - 8 = 0
so b^2 - 4ac = 9 - 4(2)(-8) = ......
4(x + 2)^2 = 36
divide both sides by 4
(x+2)^2 = 9
x+2 = ±√9
x = -2 ± 3
= -5 or 1
after looking at the above stuff, you should be able to answer your last question