x^2+8x+16=25
did you notice the left side is a perfect square?
(x+4)^2 = 25
take √ of both sides
x+4 = ± 5
x = -4 ± 5
= 1 or -9
42
To solve the equation x^2 + 8x + 16 = 25, we need to apply the quadratic formula or factor the expression on the left side.
1. Quadratic Formula Method:
The quadratic formula states that for an equation of the form ax^2 + bx + c = 0, the solutions can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
In our case, a = 1, b = 8, and c = 16 - 25 = -9. Plugging in these values, we get:
x = (-8 ± √(8^2 - 4*1*(-9))) / (2*1)
= (-8 ± √(64 + 36)) / 2
= (-8 ± √100) / 2
= (-8 ± 10) / 2
Therefore, we have two solutions:
x1 = (-8 + 10) / 2 = 2 / 2 = 1
x2 = (-8 - 10) / 2 = -18 / 2 = -9
So, the solutions to the equation x^2 + 8x + 16 = 25 are x = 1 and x = -9.
2. Factoring Method:
We can also solve the equation by factoring. Rearranging the equation, we have:
x^2 + 8x + 16 - 25 = 0
x^2 + 8x - 9 = 0
To factor this expression, we need to find two numbers whose product is -9 and whose sum is 8. These numbers are 9 and -1.
So, we can rewrite the equation as:
(x + 9)(x - 1) = 0
By applying the zero-product property, we set each factor equal to zero:
x + 9 = 0 or x - 1 = 0
Solving these equations, we get:
x = -9 or x = 1
Again, the solutions to the equation x^2 + 8x + 16 = 25 are x = -9 and x = 1.