The Quadratic Formula: Your Ultimate Tool for Solving Quadratics

A clear, practical guide for students, teachers, homeschool families, and anyone learning algebra.

Why it matters: The quadratic formula works for every quadratic equation in standard form, even when factoring is difficult or impossible with whole numbers.

The Standard Form

First write the equation as ax² + bx + c = 0, where a ≠ 0. Then identify the coefficients a, b, and c. Be careful: the signs belong to the coefficients.

The Quadratic Formula

x = (−b ± √(b² − 4ac)) ÷ (2a)

On a small phone screen, swipe horizontally inside the blue formula box if needed.

The symbol ± means “plus or minus,” so the formula usually produces two solutions: one using the plus sign and one using the minus sign.

The Discriminant Predicts the Answers

The expression under the square-root symbol, b² − 4ac, is called the discriminant. Its value tells you what kind of solutions to expect:

Discriminant	Meaning
Positive	Two different real solutions
Zero	One repeated real solution
Negative	No real solutions

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The Five-Step Process

1. Move every term to one side and write the equation as ax² + bx + c = 0.
2. Identify a, b, and c, including their signs.
3. Substitute them into the formula using parentheses.
4. Evaluate the discriminant and simplify the numerator.
5. Divide by 2a, then check the solutions in the original equation.

Worked Example 1: Two Rational Solutions

Solve: 2x² + 3x − 2 = 0

Here, a = 2, b = 3, and c = −2.

x = (−3 ± √(3² − 4 × 2 × (−2))) ÷ (2 × 2)
x = (−3 ± √25) ÷ 4 = (−3 ± 5) ÷ 4

Use both signs: x = (−3 + 5) ÷ 4 = 1 ÷ 2, or x = (−3 − 5) ÷ 4 = −2.

Solutions: x = 1 ÷ 2 and x = −2

Worked Example 2: An Irrational Answer

Solve: x² + 2x − 7 = 0

Here, a = 1, b = 2, and c = −7.

x = (−2 ± √(2² − 4 × 1 × (−7))) ÷ 2
x = (−2 ± √32) ÷ 2 = (−2 ± 4√2) ÷ 2

Divide each part of the numerator by 2:

Solutions: x = −1 + 2√2 and x = −1 − 2√2

These are real but irrational solutions because √2 cannot be written as a terminating or repeating decimal.

Worked Example 3: No Real Solutions

Analyze: 3x² + 4x + 5 = 0

The discriminant is 4² − 4 × 3 × 5 = 16 − 60 = −44. Because it is negative, the square root is not a real number.

Conclusion: the equation has no real solutions.

A Memory Tip

Opposite B, plus-or-minus, square-root the inside; divide by two A.
Say the formula rhythmically while writing it.

Common Mistakes to Avoid

• Skipping standard form: Identify coefficients only after the equation equals zero.
• Dropping a negative sign: For c = −7, substitute the entire value in parentheses.
• Forgetting ±: Calculate both the plus and minus cases.
• Using 2b in the denominator: The denominator is 2a.
• Rounding too early: Keep exact radicals or fractions until the final step.

Practice Questions

Use the quadratic formula unless a simpler method is clearly available. Give exact answers when possible.

Question	Solve
1	x2 − 6x + 8 = 0
2	2x2 − 7x + 3 = 0
3	x2 + 2x − 7 = 0
4	3x2 + 4x + 5 = 0

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Answers and Quick Checks

1. The discriminant is 4, so x = 2 or x = 4.

2. The discriminant is 25, so x = 3 or x = 1 ÷ 2.

3. The discriminant is 32, so x = −1 + 2√2 or x = −1 − 2√2.

4. The discriminant is −44, so there are no real solutions.

Final Takeaway

The quadratic formula is a dependable tool for any quadratic in standard form.

Write it as ax² + bx + c = 0, substitute carefully, use both signs, and check your result.