Quadratic Equation Solver
Solve ax² + bx + c = 0 • 2026 rates
Quadratic Formula:
Show the calculator\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
Where:
- \( a, b, c \) = Coefficients in standard form \( ax^2 + bx + c = 0 \)
- \( x \) = Variable to solve for
- \( \Delta = b^2 - 4ac \) = Discriminant
This formula finds the roots (solutions) of any quadratic equation. The discriminant determines the nature of the roots: positive for two real roots, zero for one real root, and negative for two complex roots. The formula is derived from completing the square method.
Example: For the equation \( 2x^2 + 5x - 3 = 0 \):
\( a = 2, b = 5, c = -3 \)
\( \Delta = 5^2 - 4(2)(-3) = 25 + 24 = 49 \)
\( x = \frac{-5 \pm \sqrt{49}}{2(2)} = \frac{-5 \pm 7}{4} \)
\( x_1 = \frac{2}{4} = 0.5, x_2 = \frac{-12}{4} = -3 \)
Thus, the solutions are x = 0.5 and x = -3.
Quadratic Parameters
Advanced Options
Solution Analysis
| Step | Action | Expression |
|---|---|---|
| 1 | Identify coefficients | a=1, b=-5, c=6 |
| 2 | Calculate discriminant | Δ = b² - 4ac = 25 - 24 = 1 |
| 3 | Apply quadratic formula | x = (-b ± √Δ) / (2a) |
| 4 | Calculate roots | x₁ = (5 + 1)/2 = 3, x₂ = (5 - 1)/2 = 2 |
| 5 | Verify solutions | 1(3)² - 5(3) + 6 = 0 |
| Solution | Verification | Status |
|---|---|---|
| x₁ = 3 | 1(3)² - 5(3) + 6 = 9 - 15 + 6 = 0 | ✓ Correct |
| x₂ = 2 | 1(2)² - 5(2) + 6 = 4 - 10 + 6 = 0 | ✓ Correct |
Comprehensive Quadratic Guide
A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0, where a ≠ 0. The graph of a quadratic equation is a parabola, which opens upward if a > 0 and downward if a < 0. Quadratic equations can have zero, one, or two real solutions depending on the discriminant value.
The quadratic formula is derived by completing the square on the standard form:
x² + (b/a)x + c/a = 0
x² + (b/a)x = -c/a
x² + (b/a)x + (b/2a)² = (b/2a)² - c/a
(x + b/2a)² = (b² - 4ac) / (4a²)
x = (-b ± √(b² - 4ac)) / (2a)
Where:
- a, b, c are coefficients with a ≠ 0
- Δ = b² - 4ac is the discriminant
- The ± symbol indicates two possible solutions
The discriminant (Δ = b² - 4ac) determines the nature of the roots:
- Δ > 0: Two distinct real roots
- Δ = 0: One repeated real root (perfect square)
- Δ < 0: Two complex conjugate roots
For a quadratic equation ax² + bx + c = 0:
- Vertex x-coordinate: h = -b/(2a)
- Vertex y-coordinate: k = f(h) = c - b²/(4a)
- Axis of Symmetry: x = h
- Direction: Opens up if a > 0, down if a < 0
Quadratic Learning Quiz
What is the discriminant of the equation 2x² - 4x + 2 = 0, and what does it tell us about the roots?
The answer is B) Δ = 0, one repeated real root. Using the discriminant formula: Δ = b² - 4ac = (-4)² - 4(2)(2) = 16 - 16 = 0. When the discriminant equals zero, there is exactly one real root (which is repeated).
This example demonstrates a perfect square trinomial (2x² - 4x + 2 = 2(x-1)²). When the discriminant equals zero, the quadratic touches the x-axis at exactly one point, creating a repeated root. This occurs when the quadratic can be factored as a perfect square, meaning the parabola has its vertex on the x-axis.
Discriminant: b² - 4ac, determines nature of roots
Repeated Root: Root that appears twice in solution
Perfect Square Trinomial: Quadratic that factors to (x + a)²
• Δ > 0: Two distinct real roots
• Δ = 0: One repeated real root
• Δ < 0: Two complex roots
• Remember: Δ = b² - 4ac
• Perfect squares always have Δ = 0
• Check by factoring when possible
• Confusing Δ = 0 with no real roots
• Forgetting to multiply by 4 in discriminant
• Misidentifying coefficients
Find the vertex of the parabola represented by y = x² - 6x + 8. Show your work.
For y = ax² + bx + c, the vertex x-coordinate is h = -b/(2a).
Step 1: Identify coefficients: a=1, b=-6, c=8
Step 2: Calculate x-coordinate: h = -(-6)/(2×1) = 6/2 = 3
Step 3: Calculate y-coordinate: k = f(3) = (3)² - 6(3) + 8 = 9 - 18 + 8 = -1
Therefore, the vertex is at (3, -1).
The vertex represents the minimum (if a > 0) or maximum (if a < 0) point of the parabola. The x-coordinate of the vertex is always at x = -b/(2a), which is the axis of symmetry of the parabola. The y-coordinate is found by substituting the x-coordinate back into the original equation.
Vertex: Highest or lowest point on parabola
Axis of Symmetry: Vertical line through vertex
Minimum/Maximum: Lowest/highest y-value• Vertex x-coordinate: h = -b/(2a)
• Vertex y-coordinate: k = f(h)
• Opens up if a > 0, down if a < 0
• Vertex x is always -b/(2a)
• Substitute to find y-coordinate
• Vertex is on axis of symmetry
• Forgetting the negative sign in -b
• Calculating y-coordinate incorrectly
• Confusing vertex with roots
A ball is thrown upward with an initial velocity of 32 ft/s from a height of 4 feet. Its height is modeled by h(t) = -16t² + 32t + 4. When does the ball reach its maximum height, and what is that height?
The maximum height occurs at the vertex of the parabola.
Step 1: Identify coefficients: a=-16, b=32, c=4
Step 2: Find time at vertex: t = -b/(2a) = -32/(2×-16) = -32/-32 = 1 second
Step 3: Find maximum height: h(1) = -16(1)² + 32(1) + 4 = -16 + 32 + 4 = 20 feet
Therefore, the ball reaches its maximum height of 20 feet after 1 second.
This physics application demonstrates how quadratic equations model projectile motion. The negative coefficient of the squared term indicates the parabola opens downward, which makes sense for a ball thrown upward that eventually falls back down. The vertex represents the peak of the trajectory.
Projectile Motion: Object moving under gravity
Vertex: Peak of trajectory
Initial Velocity: Starting speed and direction
• Gravity creates downward parabola (negative a)
• Maximum height at vertex
• Time and height are related quadratically
• Look for maximum/minimum keywords
• Identify vertex for optimal values
• Check if parabola opens up or down
• Not identifying the vertex as maximum
• Forgetting negative sign in gravity term
• Confusing time and height values
If a quadratic equation ax² + bx + c = 0 has roots r₁ and r₂, what is the relationship between the coefficients and the roots? Verify with the equation x² - 5x + 6 = 0.
For ax² + bx + c = 0 with roots r₁ and r₂:
• Sum of roots: r₁ + r₂ = -b/a
• Product of roots: r₁ × r₂ = c/a
For x² - 5x + 6 = 0, we have a=1, b=-5, c=6.
The equation factors as (x-2)(x-3) = 0, giving roots r₁ = 2, r₂ = 3.
Verification: r₁ + r₂ = 2 + 3 = 5 = -(-5)/1 = -b/a ✓
r₁ × r₂ = 2 × 3 = 6 = 6/1 = c/a ✓
These are Vieta's formulas, relating coefficients to roots.
Vieta's formulas establish a fundamental relationship between polynomial coefficients and their roots. For quadratics, the sum of roots equals -b/a and the product equals c/a. This relationship helps verify solutions and is foundational for understanding higher-degree polynomials.
Vieta's Formulas: Relations between coefficients and roots
Roots: Solutions to the equation
Sum/Product Relationships: Connections between coefficients and roots
• Sum of roots = -b/a
• Product of roots = c/a
• These hold for any quadratic equation
• Use to verify solutions
• Helps construct equations from roots
• Applies to higher-degree polynomials
• Forgetting the negative sign in sum formula
• Mixing up sum and product relationships
• Not accounting for coefficient a
Which of the following quadratic equations has complex roots?
The answer is C) x² + x + 1 = 0. For complex roots, the discriminant must be negative. Let's check each option:
A) x² - 4x + 4: Δ = 16 - 16 = 0 (real repeated)
B) x² - 2x - 3: Δ = 4 + 12 = 16 (two real roots)
C) x² + x + 1: Δ = 1 - 4 = -3 (complex roots)
D) x² - 9: Δ = 0 + 36 = 36 (two real roots)
Only option C has a negative discriminant, indicating complex roots.
Complex roots occur when the discriminant is negative, meaning we're taking the square root of a negative number. The solutions will be complex conjugates: x = (-b ± i√|Δ|) / (2a), where i is the imaginary unit. This happens when the parabola doesn't intersect the x-axis.
Complex Numbers: Numbers with real and imaginary parts
Imaginary Unit: i = √(-1)
Complex Conjugates: a±bi pairs
• Δ < 0 → Complex roots
• Complex roots come in conjugate pairs
• Parabola doesn't intersect x-axis
• Check discriminant first to determine root type
• Complex roots still satisfy the equation
• Graphically, parabola stays above or below x-axis
• Saying "no solutions" when complex exist
• Forgetting to include i in complex solutions
• Not recognizing conjugate pairs
Quadratic Fundamentals
x = (-b ± √(b² - 4ac)) / (2a)
Δ = b² - 4ac determines the nature of roots
Positive: Two real roots | Zero: One real root | Negative: Two complex roots
- a ≠ 0 for quadratic equations
- Always identify coefficients correctly
- Calculate discriminant first
- Consider practical constraints in applications
Advanced Quadratic Applications
y = a(x - h)² + k where (h, k) is the vertex
- Substitute solutions back into original equation
- Use discriminant to verify root nature
- Apply Vieta's formulas for verification
- Graph to visualize solutions
- Check for extraneous solutions
- Physical problems may exclude complex solutions
- Quadratic equations model parabolic relationships
- Complex roots always appear in conjugate pairs
- Always verify solutions in original equation
FAQ
Q: When should I use the quadratic formula instead of factoring?
A: Use the quadratic formula when:
- The quadratic doesn't factor easily
- Coefficients are large or complex
- The discriminant is not a perfect square
- You need exact solutions (not approximations)
- Working with complex numbers
Factoring is faster when possible, but the quadratic formula works for ALL quadratic equations. It's especially useful when the discriminant is negative (complex roots) or when factoring seems difficult.
Q: What's the difference between real and complex roots?
A: The key differences are:
Real Roots:
- Can be plotted on the real number line
- Occur when discriminant Δ ≥ 0
- Represent x-intercepts of the parabola
- Correspond to actual intersection points with x-axis
Complex Roots:
- Contain imaginary component (i = √(-1))
- Occur when discriminant Δ < 0
- Appear in conjugate pairs (a±bi)
- Parabola doesn't intersect x-axis
In practical applications, complex roots often indicate that the physical situation described by the equation cannot occur under the given conditions.