solve the quadratic equation using the square root property.
(2x+5)^2=12
2x+5 = ±√12
2x = -5 ±√12
x = (-5 ± 2√3)/2
To solve the quadratic equation using the square root property, we follow these steps:
Step 1: Rewrite the equation in the form of (ax^2 + bx + c = 0).
Given equation: (2x + 5)^2 = 12
Expanding the square: (2x + 5)(2x + 5) = 12
Simplifying: 4x^2 + 20x + 25 = 12
Step 2: Move the constant term to the right side of the equation.
4x^2 + 20x + 25 - 12 = 0
4x^2 + 20x + 13 = 0
Step 3: Apply the square root property. The square root property states that if (ax^2 + bx + c = 0), then x can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 4, b = 20, and c = 13.
x = (-20 ± √(20^2 - 4 * 4 * 13)) / (2 * 4)
x = (-20 ± √(400 - 208)) / 8
x = (-20 ± √192) / 8
Step 4: Simplify the square root (√192).
√192 can be expressed as √(64 * 3) since 64 is a perfect square.
√(64 * 3) = √64 * √3 = 8√3
Now we have:
x = (-20 ± 8√3) / 8
Step 5: Simplify further if possible.
Dividing both the numerator (-20 ± 8√3) and denominator 8 by the common factor of 8, we get:
x = (-20 ± √3) / 2
Therefore, the solutions to the quadratic equation (2x + 5)^2 = 12 using the square root property are:
x = (-20 + √3) / 2
x = (-20 - √3) / 2