Understanding the Quadratic Formula
We are going to explore the quadratic formula, a powerful tool for solving quadratic equations. A quadratic equation is an equation of the form:
ax² + bx + c = 0
where a, b, and c are constants, and a is not equal to 0.
What is the Quadratic Formula?
The quadratic formula is:
x = (-b ± √(b²-4ac)) / (2a)
This formula allows us to find the roots of any quadratic equation.
Steps to Use the Quadratic Formula:
- Identify the coefficients a, b, and c in your quadratic equation.
- Plug these values into the quadratic formula.
- Simplify under the square root, known as the discriminant.
- Determine the two solutions by using both the plus and minus signs.
- Simplify the expressions to find the roots of the equation.
Example:
Solve the quadratic equation 2x² – 4x – 6 = 0 using the quadratic formula.
-
- First, identify the coefficients: a=2, b=-4, c=-6.
- Plug them into the formula:
x = (-(-4) ± √((-4)²-4*2*(-6))) / (2*2)
-
- Simplify the discriminant (√(16+48)):
x = (4 ± √64) / 4
-
- Solve for both possible values of x:
x = (4 ± 8) / 4
x = (4 + 8) / 4 or x = (4 – 8) / 4
-
- Thus, the solutions are:
x = 3 or x = -1
Common Questions:
-
- What if the discriminant is zero?
If the discriminant (the part under the square root) is zero, the equation has one real root.
-
- What if the discriminant is negative?
If the discriminant is negative, the equation has two complex roots.
-
- Why do we divide by 2a?
We divide by 2a to solve for x. This is derived from the process of completing the square in the quadratic equation.
Practice Problems:
Solve the following quadratic equations using the quadratic formula:
- x² + 6x + 9 = 0
- 3x² – 2x – 8 = 0
- x² – 4x + 4 = 0
Let’s solve the first problem together:
-
- Identify a=1, b=6, c=9.
- Use the quadratic formula:
x = (-6 ± √(6²-4*1*9)) / (2*1)
-
- Simplify the discriminant:
x = (-6 ± √(36-36)) / 2
x = (-6 ± 0) / 2
-
- Since the discriminant is zero, there is one real root:
x = -6 / 2
-
- The solution is:
x = -3
Remember: Practice is key to understanding the quadratic formula!
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