Solve the following quadratic equation using square roots. Round to the nearest hundredth if necessary: (x−6)2=51
To solve the quadratic equation, we will first take the square root of both sides of the equation:
√((x-6)^2) = √51
Simplifying the left side:
x - 6 = ±√51
Next, we will isolate x by adding 6 to both sides:
x = 6 ± √51
Rounding to the nearest hundredth:
x ≈ 6 + 7.14 ≈ 13.14 or x ≈ 6 - 7.14 ≈ -1.14
Therefore, the solutions to the quadratic equation (x-6)^2 = 51 are approximately x = 13.14 and x = -1.14.
To solve the quadratic equation (x−6)²=51, we need to isolate the variable x and then take the square root of both sides.
Step 1: Expand the equation:
(x−6)² = 51
x² - 12x + 36 = 51
Step 2: Move the constant term to the other side:
x² - 12x + 36 - 51 = 0
x² - 12x - 15 = 0
Step 3: Use the formula x = (-b ± √(b² - 4ac)) / (2a) to find the solutions.
In this case, a = 1, b = -12, and c = -15.
x = (-(-12) ± √((-12)² - 4(1)(-15))) / (2(1))
x = (12 ± √(144 + 60)) / 2
x = (12 ± √204) / 2
x = (12 ± 2√51) / 2
x = 6 ± √51
So, the solutions to the quadratic equation are x = 6 + √51 and x = 6 - √51.
To solve the quadratic equation (x−6)² = 51 using square roots, follow these steps:
Step 1: Expand the equation
(x−6)² = 51
(x−6)(x−6) = 51
x² − 12x + 36 = 51
Step 2: Move the constant term to the other side
x² − 12x + 36 - 51 = 0
x² − 12x - 15 = 0
Step 3: Use the quadratic formula to find x
The quadratic formula is given by:
x = (-b ± √(b² - 4ac)) / (2a)
In our equation, a = 1, b = -12, and c = -15. Plugging these values into the quadratic formula, we get:
x = (-(-12) ± √((-12)² - 4(1)(-15))) / (2(1))
Simplifying further:
x = (12 ± √(144 + 60)) / 2
x = (12 ± √204) / 2
Step 4: Simplify the square root term
√204 can be simplified as follows:
√204 = √(4 * 51)
√204 = 2√51
Now, substitute 2√51 back into the equation:
x = (12 ± 2√51) / 2
Step 5: Simplify and find the solutions
x = (12 ± 2√51) / 2
x = 6 ± √51
The two solutions to the quadratic equation are:
x = 6 + √51
x = 6 - √51
Rounding to the nearest hundredth (two decimal places):
x ≈ 11.18 or -5.18