📖Definition
The quadratic formula is a mathematical formula that provides the solution(s) to a quadratic equation of the form ax² + bx + c = 0. It gives you the values of x that make the equation true.
📐Formula
In this formula: a is the coefficient of x², b is the coefficient of x, c is the constant term, and the ± symbol means you calculate two solutions (one with + and one with -).
🔎What the discriminant tells you
Two real solutions
If b² − 4ac is positive, the square root is real and the equation has two distinct real solutions.
One real solution
If b² − 4ac equals zero, the two formula results are the same, giving one repeated real solution.
No real solutions
If b² − 4ac is negative, the square root is not real, so there are no real-number solutions.
📝Step-by-Step Guide
Identify the coefficients
Write your equation in standard form (ax² + bx + c = 0) and identify the values of a, b, and c.
Calculate the discriminant
The discriminant tells you how many solutions exist. Calculate b² - 4ac.
Apply the formula
Substitute your values into the quadratic formula and simplify.
Calculate both solutions
Use the + sign for one solution and the - sign for the other.
⚠️Common Mistakes to Avoid
- Forgetting to include the negative sign in front of b
- Making errors when calculating the discriminant (b² - 4ac)
- Dividing only part of the numerator by 2a instead of the entire expression
- Not simplifying the square root when possible
- Forgetting that ± gives two different answers
✏️Practice Problems
Solve x² - 5x + 6 = 0
💡 Hint: Identify a=1, b=-5, c=6
Answer: x = 2 or x = 3
Solve 2x² + 7x - 15 = 0
💡 Hint: Remember to use a=2 when dividing
Answer: x = 1.5 or x = -5
Solve 3x² - 2x - 8 = 0
Answer: x = 2 or x = -4/3
Solve x² + 4x + 4 = 0
Answer: x = -2
Identify a, b, and c in 2x² - 3x + 1 = 0
Answer: a = 2, b = -3, c = 1
Solve x² - 2x - 8 = 0 using the quadratic formula
Answer: x = 4 or x = -2
Find the discriminant of 2x² + 4x + 5 = 0
Answer: Δ = -24
Solve 4x² + 4x - 3 = 0
Answer: x = 1/2 or x = -3/2
For what value of k does x² + 6x + k = 0 have one real solution?
Answer: k = 9
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