The method for completing the square can be used to write the expression -2x^2+12x-5 in the form a(x+b)^2 +c, where a,b, and c are constants. What is the value of c? Thank you for any help.
-2x^2+12x-5
-2(x^2-6x +9) -5 -2*9
-2(x-3)^2 -23
check that.
To write the expression -2x^2 + 12x - 5 in the form a(x+b)^2 + c, where a, b, and c are constants, we need to complete the square.
Step 1: Group the terms (-2x^2 + 12x) together: -2(x^2 - 6x).
Step 2: To complete the square, take half of the coefficient of the x-term, square it, and add it to both sides of the equation.
Half of the coefficient of the x-term is (1/2) * 12 = 6. Squaring it gives 6^2 = 36.
So we add 36 to both sides:
-2(x^2 - 6x) + 36 = -5 + 36
Step 3: Simplify both sides of the equation:
-2(x^2 - 6x) + 36 = 31
Step 4: Factor the trinomial inside the parentheses:
(x - 3)^2 = 31/(-2) (Dividing both sides by -2)
Step 5: Rewrite the equation in the desired form:
(x - 3)^2 = -31/2
Step 6: Comparing it to the form a(x + b)^2 + c, we can see that a = -1/2, b = -3, and c = -31/2.
Therefore, the value of c is -31/2.
To complete the square, we first need to factor out the common factor in the quadratic equation:
-2x^2 + 12x - 5
= -2(x^2 - 6x) - 5
Next, we need to find the constant term needed to complete the square. To do this, we take half the coefficient of the x-term and square it.
The coefficient of the x-term is 6, so half of it is 6/2 = 3. Squaring 3, we get 9.
So, we rewrite the expression as:
-2(x^2 - 6x + 9 - 9) - 5
= -2(x^2 - 6x + 9) + 18 - 5
= -2(x - 3)^2 + 13
Now we can see that the expression is in the form a(x + b)^2 + c, where a = -2, b = 3, and c = 13.
Therefore, the value of c is 13.