Enter your answer and show all the steps that you use to solve this problem in the space provided.
What are the solutions of the quadratic equation?
2x^2 - 16x + 32 = 0
To find the solutions of the quadratic equation 2x^2 - 16x + 32 = 0, we can use the quadratic formula.
The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions for x are given by:
x = (-b ± sqrt(b^2 - 4ac)) / 2a
For our equation 2x^2 - 16x + 32 = 0, we have a = 2, b = -16, and c = 32.
Plugging these values into the quadratic formula, we get:
x = (-(-16) ± sqrt((-16)^2 - 4(2)(32))) / (2(2))
= (16 ± sqrt(256 - 256)) / 4
= (16 ± sqrt(0)) / 4
Since the discriminant, which is the term inside the square root, is equal to 0, we have a unique solution. This means that the quadratic equation has one real root.
Simplifying further, we have:
x = (16 ± 0) / 4
= 16/4
= 4
Therefore, the solution to the quadratic equation 2x^2 - 16x + 32 = 0 is x = 4.
To find the solutions of the quadratic equation 2x^2 - 16x + 32 = 0, we can use either factoring, completing the square, or the quadratic formula.
Method 1: Factoring
Step 1: Set the equation equal to zero: 2x^2 - 16x + 32 = 0.
Step 2: Divide each term by 2 to simplify the equation: x^2 - 8x + 16 = 0.
Step 3: Factorize the quadratic equation: (x - 4)(x - 4) = 0.
Step 4: Set each factor equal to zero and solve for x:
x - 4 = 0.
Step 5: Solve for x:
x = 4.
Method 2: Completing the square
Step 1: Set the equation equal to zero: 2x^2 - 16x + 32 = 0.
Step 2: Divide each term by 2 to simplify the equation: x^2 - 8x + 16 = 0.
Step 3: Take 16 from both sides of the equation: x^2 - 8x = -16.
Step 4: Complete the square on the left side of the equation:
(x - 4)^2 = 0.
Step 5: Take the square root of both sides:
x - 4 = 0.
Step 6: Solve for x:
x = 4.
Method 3: Quadratic formula
Step 1: Identify the coefficients of the quadratic equation:
a = 2, b = -16, c = 32.
Step 2: Substitute the values into the quadratic formula:
x = (-b ± √(b^2 - 4ac)) / (2a).
Step 3: Simplify the equation:
x = (-(-16) ± √((-16)^2 - 4(2)(32))) / (2(2)).
Step 4: Solve the equation:
x = (16 ± √(256 - 256)) / 4.
Step 5: Simplify further:
x = (16 ± √0) / 4.
Step 6: Solve for x:
x = (16 ± 0) / 4.
Step 7: Simplify the expression:
x = 16/4 = 4.
Therefore, the solution to the quadratic equation 2x^2 - 16x + 32 = 0 is x = 4.
To find the solutions of the quadratic equation 2x^2 - 16x + 32 = 0, we can use the quadratic formula. The quadratic formula states that for an equation in the form ax^2 + bx + c = 0, the solutions are given by:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our equation, a = 2, b = -16, and c = 32. Substituting these values into the quadratic formula, we get:
x = (-(-16) ± √((-16)^2 - 4 * 2 * 32)) / (2 * 2)
x = (16 ± √(256 - 256)) / 4
x = (16 ± √0) / 4
Since the square root of 0 is 0, we have:
x = (16 ± 0) / 4
x = 16/4
x = 4
Therefore, the quadratic equation 2x^2 - 16x + 32 = 0 has a single solution x = 4.
Note: In this case, there is no need to go through the entire quadratic formula because the discriminant (b^2 - 4ac) equals zero, indicating that there is only one real solution.