Consider the equation: x^2+2x-15=0
A) First, use the "completing the square" process to write this equation in the form (x+D)^2=E and enter your results below.
x^2+2x-15=0 is equivalent to:
Preview left side of eqn:
B) Solve your equation and enter your answers below as a list of numbers, separated with a comma where necessary.
A) To complete the square, we need to add a constant term on both sides of the equation. We can do this by adding and subtracting the square of half of the coefficient of x from both sides.
x^2+2x-15=0
Taking the coefficient of x, which is 2. Half of 2 is 1, and its square is 1^2 = 1.
Adding and subtracting 1 on the left side:
x^2+2x+1-1-15=0
Now, we can rewrite the left side as a perfect square trinomial:
(x+1)^2-16=0
So, the equation x^2+2x-15=0 is equivalent to (x+1)^2-16=0.
Preview left side of eqn: (x+1)^2
B) To solve the equation (x+1)^2-16=0, we can set the equation equal to zero and factor the left side:
(x+1)^2-16=0
(x+1)^2=16
Taking the square root of both sides:
x+1=±√16
x+1=±4
Subtracting 1 from both sides:
x=-1±4
Simplifying:
x=-1+4, x=-1-4
x=3, x=-5
So, the solution to the equation x^2+2x-15=0 is x=3 and x=-5.
A) To complete the square, we need to add a constant term to both sides of the equation.
First, let's move the constant term to the right side of the equation:
x^2 + 2x = 15
Next, we take half of the coefficient of the x term (which is 2) and square it:
(2/2)^2 = 1^2 = 1
Then, we add this constant term to both sides of the equation:
x^2 + 2x + 1 = 15 + 1
x^2 + 2x + 1 = 16
Now, we can write the left side of the equation as a perfect square:
(x + 1)^2 = 16
So, the equation x^2 + 2x - 15 = 0 is equivalent to (x + 1)^2 = 16.
B) To solve the equation (x + 1)^2 = 16, we can take the square root of both sides:
√((x + 1)^2) = ±√16
(x + 1) = ±4
Next, we isolate x by subtracting 1 from both sides:
x + 1 - 1 = ±4 - 1
x = -1 ± 4
Therefore, the solutions to the equation x^2 + 2x - 15 = 0 are x = -1 + 4 and x = -1 - 4.
So, the solutions are x = 3 and x = -5.