solve the equation using the square root properly.
x^2-6x+9=16
The given equation is x^2 - 6x + 9 = 16.
First, let's rearrange the terms to form a quadratic equation:
x^2 - 6x + 9 - 16 = 0
which simplifies to:
x^2 - 6x - 7 = 0
Next, let's use the quadratic formula to solve for x. The quadratic formula is given by:
x = [-b ± sqrt(b^2 - 4ac)] / (2a)
where a, b, and c are the coefficients from the quadratic equation ax^2 + bx + c = 0.
Here, a = 1, b = -6, and c = -7
Let's substitute these values into the quadratic formula to solve for x:
x = [6 ± sqrt((-6)^2 - 4*1*(-7))] / (2*1)
which simplifies to:
x = [6 ± sqrt(36 + 28)] / 2
Further simplifying yields:
x = [6 ± sqrt(64)] / 2
Taking the square root of 64 results in 8. Therefore, we will have two solutions:
x = (6 + 8) / 2 = 14 / 2 = 7
and
x = (6 - 8) / 2 = -2 / 2 = -1
Therefore, the solutions to the equation are x = 7 and x = -1.
To solve the equation x^2 - 6x + 9 = 16 using the square root method, follow these steps:
Step 1: Move all terms to one side of the equation to obtain a quadratic equation set equal to zero.
x^2 - 6x + 9 - 16 = 0
Simplifying, we get:
x^2 - 6x - 7 = 0
Step 2: Apply the square root property, which states that if x^2 = a, then x = ±√a.
To do this, rearrange the equation and isolate the x^2 term:
x^2 - 6x - 7 = 0
Step 3: Take the square root of both sides of the equation and solve for x.
Taking the square root of both sides, we have:
√(x^2 - 6x - 7) = ±√0
Simplifying the right side, we get:
√(x^2 - 6x - 7) = 0
Step 4: Solve for x by isolating x in each equation formed by the square root property.
Solving for x, we have:
x = ±√(6x + 7)
Step 5: Solve for x using the positive and negative square roots.
Consider the positive square root:
x = √(6x + 7)
Square both sides to eliminate the square root:
x^2 = 6x + 7
Rearrange the equation and set it equal to zero:
x^2 - 6x - 7 = 0
Now consider the negative square root:
x = -√(6x + 7)
Square both sides to eliminate the square root:
x^2 = 6x + 7
Rearrange the equation and set it equal to zero:
x^2 - 6x - 7 = 0
Both equations are the same, so we only need to solve one of them.
Step 6: Solve the quadratic equation by factoring, completing the square, or using the quadratic formula.
By factoring or other methods, we can find that the equation factors as:
(x - 7)(x + 1) = 0
So, either (x - 7) = 0 or (x + 1) = 0
Solving each equation separately, we have:
For (x - 7) = 0, x = 7
For (x + 1) = 0, x = -1
Step 7: Check for extraneous solutions.
Plug both x = 7 and x = -1 back into the original equation:
x^2 - 6x + 9 = 16
For x = 7:
(7)^2 - 6(7) + 9 = 49 - 42 + 9 = 16 (satisfied)
For x = -1:
(-1)^2 - 6(-1) + 9 = 1 + 6 + 9 = 16 (satisfied)
Therefore, the solutions to the equation x^2 - 6x + 9 = 16 are x = 7 and x = -1.
To solve the equation x^2 - 6x + 9 = 16 using the square root, follow these steps:
Step 1: Move all the terms to one side to have a zero on the other side of the equation:
x^2 - 6x + 9 - 16 = 0
Simplify by combining like terms:
x^2 - 6x - 7 = 0
Step 2: Apply the quadratic formula, which states that the solutions to a quadratic equation of the form ax^2 + bx + c = 0 can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For the given equation, the coefficients are:
a = 1, b = -6, and c = -7
Substitute these values into the quadratic formula:
x = (-(-6) ± √((-6)^2 - 4(1)(-7))) / (2(1))
Simplify:
x = (6 ± √(36 + 28)) / 2
x = (6 ± √64) / 2
x = (6 ± 8) / 2
Step 3: Simplify further:
For the positive root:
x1 = (6 + 8) / 2
x1 = 14 / 2
x1 = 7
For the negative root:
x2 = (6 - 8) / 2
x2 = -2 / 2
x2 = -1
Therefore, the solutions to the equation x^2 - 6x + 9 = 16 are x = 7 and x = -1.