Complete the square to solve the following quadratic equation: x2+2x−8=0 .(1 point) x equals 4 x=−10, x=8 x equals negative 10 , x equals 8 x=10, x=−8 x equals 10 , x equals negative 8 x=2, x=−4
To complete the square for the quadratic equation x^2 + 2x - 8 = 0, we can rearrange the equation as follows:
x^2 + 2x = 8
To create a perfect square trinomial on the left side of the equation, we need to take half of the coefficient of x, square it, and add it to both sides of the equation.
The coefficient of x is 2, so half of it is 1 and its square is 1^2 = 1.
Adding 1 to both sides of the equation, we get:
x^2 + 2x + 1 = 8 + 1
Simplifying, we have:
(x + 1)^2 = 9
Now, we can take the square root of both sides of the equation:
√((x + 1)^2) = ±√9
Simplifying further:
x + 1 = ±3
Subtracting 1 from both sides of the equation, we have:
x = -1 ± 3
So, the solutions to the quadratic equation x^2 + 2x - 8 = 0 are:
x = -1 + 3 = 2
x = -1 - 3 = -4
Therefore, x equals 2 and x equals -4.
Using the quadratic formula, find the solution to x2−6x−16=0 .(1 point) Responses x=−2 x equals negative 2 x=8, x=−8 x equals 8 , x equals negative 8 x=8 x equals 8 x=8, x=−2
The quadratic formula is given by:
x = (-b ± √(b^2 - 4ac))/(2a)
For the quadratic equation x^2 - 6x - 16 = 0, the coefficients are:
a = 1
b = -6
c = -16
Plugging these values into the quadratic formula, we get:
x = (-(-6) ± √((-6)^2 - 4(1)(-16)))/(2(1))
Simplifying further:
x = (6 ± √(36 + 64))/(2)
x = (6 ± √(100))/(2)
Taking the square root:
x = (6 ± 10)/(2)
Now, we can evaluate the two possible solutions:
x_1 = (6 + 10)/(2) = 16/2 = 8
x_2 = (6 - 10)/(2) = -4/2 = -2
Therefore, the solutions to the quadratic equation x^2 - 6x - 16 = 0 are x = 8 and x = -2.
Which of the following is a solution to the quadratic equation 2x2−5x=6 ? Assume that the solution has been rounded to the nearest hundredth, if applicable.(1 point) Responses x=−1.03 x equals negative 1.03 x=1.03 x equals 1.03 x=0.89 x equals 0.89 x=−0.89
To find which of the given options is a solution to the quadratic equation 2x^2 - 5x = 6, we can substitute each option back into the equation and check if it satisfies the equation.
Option 1: x = -1.03
Substituting into the equation,
2(-1.03)^2 - 5(-1.03) = 6
2(1.0599) + 5.15 ≠ 6
Option 2: x = 1.03
Substituting into the equation,
2(1.03)^2 - 5(1.03) = 6
2(1.0599) - 5.15 ≠ 6
Option 3: x = 0.89
Substituting into the equation,
2(0.89)^2 - 5(0.89) = 6
2(0.7921) - 4.45 ≠ 6
Option 4: x = -0.89
Substituting into the equation,
2(-0.89)^2 - 5(-0.89) = 6
2(0.7921) + 4.45 ≠ 6
None of the given options satisfy the equation 2x^2 - 5x = 6.
How many real solutions does the following quadratic equation have? 4x2+x+3=0 (1 point) Responses two real solutions two real solutions one real solution one real solution three real solutions three real solutions no real solutions
To determine the number of real solutions of the quadratic equation 4x^2 + x + 3 = 0, we can use the discriminant. The discriminant is given by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.
For this equation, a = 4, b = 1, and c = 3.
Plugging these values into the discriminant formula, we have:
Discriminant = (1)^2 - 4(4)(3)
= 1 - 48
= -47
Since the discriminant is negative (-47), it means that there are no real solutions to the equation. Therefore, the quadratic equation 4x^2 + x + 3 = 0 has no real solutions.