Solve the following quadratic equation using the perfect square trinomial pattern: x^2+x+0.25=0
To solve the quadratic equation x^2 + x + 0.25 = 0 using the perfect square trinomial pattern, we need to rewrite the equation in the form (x + a)^2 = 0.
Comparing the equation with the pattern, we can observe that:
1. The coefficient of x^2 is 1, so the perfect square trinomial must have a coefficient of x^2 as 1.
2. The coefficient of x is 1, so the perfect square trinomial must have a coefficient of x as 2(a)(1) = 2(a) = 1, giving us a = 0.5.
3. We observe that the constant term is 0.25, which is the square of (a)^2 = (0.5)^2 = 0.25.
Hence, we can rewrite the quadratic equation as:
(x + 0.5)^2 = 0
To solve the equation, we take the square root of both sides:
√[(x + 0.5)^2] = √[0]
(x + 0.5) = 0
Solving for x, we get:
x = -0.5
Therefore, the solution to the quadratic equation x^2 + x + 0.25 = 0 is x = -0.5.
To solve the quadratic equation x^2 + x + 0.25 = 0 using the perfect square trinomial pattern, we can follow these steps:
Step 1: Identify the coefficients a, b, and c of the quadratic equation.
In this case, a = 1, b = 1, and c = 0.25.
Step 2: Check if the quadratic equation can be factored using the perfect square trinomial pattern.
The perfect square trinomial pattern applies when the quadratic equation is of the form (x + m)^2 = 0 or (x - m)^2 = 0.
In our equation, if we take the square root of the constant term (0.25), we get 0.5. Therefore, it gives us the hint that our equation is a perfect square trinomial since the square root of the constant term matches the coefficient of x in the middle term.
Step 3: Rewrite the quadratic equation using the perfect square trinomial pattern.
Based on the perfect square trinomial pattern, we can rewrite the equation (x + m)^2 = 0, where m = 0.5.
(x + 0.5)^2 = 0
Step 4: Solve for x.
To find x, take the square root of both sides of the equation, remembering to consider both the positive and negative square roots:
√(x + 0.5)^2 = √0
x + 0.5 = 0 or x + 0.5 = 0
Solving each equation separately:
For x + 0.5 = 0:
x = -0.5
For x + 0.5 = 0:
x = -0.5
Therefore, the solution to the quadratic equation x^2 + x + 0.25 = 0 is x = -0.5.
To solve the given quadratic equation x^2 + x + 0.25 = 0, we can use the perfect square trinomial pattern.
The perfect square trinomial pattern is:
a^2 + 2ab + b^2 = (a + b)^2
In our equation, x^2 + x + 0.25, we notice that the first two terms form a perfect square trinomial, since (x)^2 + (x)(1) = x^2 + x.
We can rewrite our equation as:
(x + 0.5)^2 = 0
Now, take the square root of both sides to isolate x:
sqrt((x + 0.5)^2) = sqrt(0)
x + 0.5 = 0
Solving for x, we have:
x = -0.5
So, the solution to the quadratic equation x^2 + x + 0.25 = 0 is x = -0.5.