8.5The Quadratic Formula
The universal solver — x = (−b ± √(b²−4ac)) / (2a) works on every quadratic equation, factorable or not, and the discriminant tells you exactly how many solutions to expect before you even start.
Why This Matters
The quadratic formula solves any quadratic equation — no matter how complex. It's one of the most important formulas in all of mathematics, used in Physics for projectile motion, in engineering for structural analysis, and on every standardized test.
Workbook
Lesson, vocabulary, worked examples, and practice problems.
Essential Question
Why is the quadratic formula a universal method, and how does the discriminant predict the number and type of solutions before you solve?
Lesson Overview
The quadratic formula x = (−b ± √(b²−4ac)) / (2a) solves any quadratic equation in standard form — no matter whether it factors nicely or not. The expression under the radical, b²−4ac, is called the discriminant. It tells you whether the equation has two real solutions, one repeated solution, or no real solutions before you do any arithmetic. Mastering this formula means you always have a reliable path to the answer, even when factoring fails.
Worked Examples
Solve x² + 5x + 6 = 0 using the quadratic formula
a=1, b=5, c=6
D = 25−24 = 1
x = (−5 ± √1) / 2 = (−5 ± 1) / 2
x = (−5+1)/2 = −2 or x = (−5−1)/2 = −3
Solve 2x² − 4x − 6 = 0
a=2, b=−4, c=−6
D = 16+48 = 64
x = (4 ± √64) / 4 = (4 ± 8) / 4
x = 12/4 = 3 or x = −4/4 = −1
Solve x² − 4x + 1 = 0 (irrational roots)
a=1, b=−4, c=1
D = 16−4 = 12
√12 = 2√3
x = (4 ± 2√3) / 2 = 2 ± √3
Solve x² − 6x + 9 = 0 (double root)
a=1, b=−6, c=9
D = 36−36 = 0
x = 6 / 2 = 3
Solve x² + 2x + 5 = 0 — determine number of solutions
a=1, b=2, c=5
D = 4−20 = −16
D < 0 → no real solutions
Guided Practice
Guided Practice Video: Quadratic Formula
Review applying the quadratic formula with multiple worked examples before completing the guided problems below.
Video by Sang Real Math
Watch on YouTube ↗Solve x² − 7x + 10 = 0 using the quadratic formula
Hint: a=1, b=−7, c=10. Compute D=49−40=9. Then x=(7±3)/2.
Solve 3x² + 6x − 9 = 0
Hint: Divide through by 3 first to simplify: x²+2x−3=0. Then apply the formula.
Use the discriminant to determine the number of solutions: x² + 4x + 5 = 0
Hint: Compute D=b²−4ac=16−20. What is the sign of D?
Solve x² − 2x − 4 = 0. Leave answer in exact form.
Hint: D=4+16=20. Simplify √20=2√5. Then x=(2±2√5)/2=1±√5.
Solve 2x² + 3x − 2 = 0
Hint: a=2, b=3, c=−2. D=9+16=25. x=(−3±5)/4.
Key Vocabulary
Quadratic Formula
x = (−b ± √(b²−4ac)) / (2a). Solves any quadratic equation ax²+bx+c=0.
Discriminant
The expression b²−4ac inside the radical. Its sign determines the number of real solutions.
Irrational Root
A solution that cannot be written as a simple fraction. Appears when the discriminant is positive but not a perfect square.
Exact Form
An answer left as a simplified radical (e.g. (3+√5)/2) rather than a decimal approximation.
Double Root
When the discriminant equals zero, both solutions are identical: x = −b/(2a).
No Real Solutions
When the discriminant is negative, the parabola does not cross the x-axis and the equation has no real roots.
Practice Questions
Interactive Practice — 5 Questions
What is the quadratic formula?
For x² − 5x + 6 = 0, what is the discriminant (b²−4ac)?
If the discriminant is negative, how many real solutions does the equation have?
Solve x² + 2x − 8 = 0 using the quadratic formula.
For 2x² − 4x + 2 = 0, what does the discriminant tell you?
Independent Practice
Independent Practice
Solve using the quadratic formula: x² + 3x − 4 = 0
Solve: x² − 5x + 6 = 0
Solve: 2x² + x − 3 = 0
Solve: x² + 6x + 9 = 0
Use the discriminant only — state the number of solutions: x² + 3x + 5 = 0
Common Mistakes
Forgetting the ± in the quadratic formula — writing only the + or only the − case.
The formula gives two solutions: x = (−b + √discriminant) / 2a AND x = (−b − √discriminant) / 2a.
Computing −b incorrectly — e.g., for b = −5, writing −b = −5 instead of +5.
−b means the opposite of b. If b = −5, then −b = +5.
Dividing only the numerator's first term by 2a — e.g., writing (−b ± √discriminant) / 2a as −b/2a ± √discriminant.
The entire numerator (−b ± √discriminant) is divided by 2a. Use parentheses to keep it together.
Forgetting to set the equation equal to zero before identifying a, b, and c.
The quadratic formula requires standard form ax² + bx + c = 0. Rearrange first if needed.
Math Tips
Memorize the quadratic formula: x = (−b ± √(b²−4ac)) / (2a). It works for every quadratic equation.
Always identify a, b, c from standard form ax²+bx+c=0 before plugging in — sign errors are the #1 mistake.
Check the discriminant first: b²−4ac > 0 → 2 solutions; = 0 → 1 solution; < 0 → no real solutions.
SAT tip: the quadratic formula always works — use it when factoring is not obvious or when the discriminant is not a perfect square.