Solve the quadratic equation by completing the square. (-x^2 + 6x + 10 = 0)
To solve the quadratic equation (-x^2 + 6x + 10 = 0) by completing the square, follow these steps:
Step 1: Move the constant term to the other side of the equation:
-x^2 + 6x = -10
Step 2: Rearrange the equation so that the x terms are grouped together:
-x^2 + 6x + 0 = -10
Step 3: To complete the square, take half of the coefficient of the x term (6/2 = 3), square it (3^2 = 9), and add it to both sides of the equation:
-x^2 + 6x + 9 = -10 + 9
-x^2 + 6x + 9 = -1
Step 4: Rewrite the left side of the equation as a perfect square trinomial. To do this, write it as (x - a)^2, where 'a' is half of the coefficient of the x term (3):
-(x^2 - 6x + 9) = -1
Step 5: Simplify the equation:
-(x - 3)^2 = -1
Step 6: Multiply both sides of the equation by -1 to eliminate the negative sign on the left side:
(x - 3)^2 = 1
Step 7: Take the square root of both sides:
√((x - 3)^2) = √1
Step 8: Solve for x by considering both the positive and negative square root:
x - 3 = ±1
Step 9: Add 3 to both sides of the equation:
x = 3 ± 1
Step 10: Simplify the expression:
x = 4 or x = 2
So, the solutions to the quadratic equation (-x^2 + 6x + 10 = 0) by completing the square are x = 4 and x = 2.
To solve the quadratic equation by completing the square, follow these steps:
Step 1: Set the equation equal to zero:
-x^2 + 6x + 10 = 0
Step 2: Move the constant term to the other side of the equation:
-x^2 + 6x = -10
Step 3: Divide the coefficient of x by 2 and square the result. Add this value to both sides of the equation to complete the square. We divide 6 by 2 to get 3, and then square it to get 9:
-x^2 + 6x + 9 = -10 + 9
-x^2 + 6x + 9 = -1
Step 4: Now, rewrite the left side of the equation as a perfect square:
-(x^2 - 6x + 9) = -1
Step 5: Factor the perfect square on the left side of the equation:
-(x - 3)^2 = -1
Step 6: Multiply both sides of the equation by -1 to eliminate the negative sign:
(x - 3)^2 = 1
Step 7: Take the square root of both sides of the equation:
x - 3 = ± √1
Step 8: Simplify the square root:
x - 3 = ± 1
Step 9: Solve for x by isolating it on one side of the equation:
Case 1: x - 3 = 1
x = 1 + 3
x = 4
Case 2: x - 3 = -1
x = -1 + 3
x = 2
So the solutions to the quadratic equation -x^2 + 6x + 10 = 0 are x = 4 and x = 2.
-x^2 + 6x + 10 = 0
-(x^2 - 6x) + 10 = 0
-(x^2 - 6x + 9) + 10 + 9 = 0
-(x-3)^2 + 19 = 0
(x-3)^2 = 19
x-3 = ±√19
x = 3±√19