Answer:
the solutions to the quadratic equation 2x^2 + 5x + 3 = 0 are x = -3 and x = -1.
Explanation:
The quadratic formula is derived using the method of completing the square. To do this, we start with the quadratic equation in standard form:
ax^2 + bx + c = 0
We divide both sides of the equation by a to get:
x^2 + (b/a)x + (c/a) = 0
We want to move the constant term to the right side of the equation, so we subtract c/a from both sides:
x^2 + (b/a)x = -(c/a)
Now, we take the following steps to complete the square:
We take half of the coefficient of the x term, square it, and add it to both sides of the equation. The coefficient of the x term is b/a, so half of it is b/2a and squaring it gives us b^2/4a^2.
We rewrite the left side of the equation as a squared term.
x^2 + (b/a)x + b^2/4a^2 = -(c/a) + b^2/4a^2
(x + b/2a)^2 = b^2/4a^2 - c/a
We can now take the square root of both sides to isolate x:
√(x + b/2a)^2 = √(b^2/4a^2 - c/a)
x + b/2a = ±√(b^2/4a^2 - c/a)
Finally, we subtract b/2a from both sides to get the quadratic formula:
x = -b/2a ±√(b^2/4a^2 - c/a)
This is the quadratic formula in its most common form. It can be used to solve any quadratic equation, regardless of the values of a, b, and c.
Here is an example of how to use the quadratic formula to solve a quadratic equation:
Solve for x: 2x^2 + 5x + 3 = 0
Step 1: Identify the values of a, b, and c.
a = 2, b = 5, c = 3
Step 2: Substitute the values of a, b, and c into the quadratic formula.
x = (-5 ±√(5^2 - 4 * 2 * 3)) / (2 * 2)
Step 3: Evaluate the quadratic formula.
x = (-5 ±√1) / 4
Step 4: Simplify the answer.
x = (-5 ±1) / 4
x = -3 or x = -1
Therefore, the solutions to the quadratic equation 2x^2 + 5x + 3 = 0 are x = -3 and x = -1.