Quadratic Formula Calculator
Use this free quadratic formula calculator to solve any quadratic equation. Enter your coefficients and get both solutions with step-by-step work, discriminant analysis, and vertex coordinates. Perfect for algebra students.
Quadratic solver
Solve a quadratic equation
Enter a, b, and c. You’ll get exact steps, decimal roots, the discriminant, vertex, and a responsive parabola graph.
Standard form: ax² + bx + c = 0
Solution
Roots
Method
Step-by-step solution
Visual check
Parabola graph
Equation facts
Properties at a glance
What is the Quadratic Formula?
The quadratic formula solves any equation of the form \( ax^2 + bx + c = 0 \), where \( a \neq 0 \).
$$x = \frac{-b \pm \sqrt{b^2 – 4ac}}{2a}$$
This formula always works, regardless of whether the equation factors nicely.
The Discriminant
The discriminant (\( b^2 – 4ac \)) determines the nature of the solutions:
| Discriminant | Solutions |
|---|---|
| Positive | Two distinct real roots |
| Zero | One repeated real root |
| Negative | Two complex conjugate roots |
Work through examples by hand before relying on calculators. Understanding the process helps you catch errors that tools miss.
Standard Form
A quadratic equation must be in standard form before applying the formula:
$$ax^2 + bx + c = 0$$
If your equation isn’t in this form, rearrange it first by moving all terms to one side.
Step-by-Step Process
- Identify coefficients – Find \( a \), \( b \), and \( c \) from your equation
- Calculate the discriminant – \( b^2 – 4ac \)
- Apply the formula – Substitute into \( \frac{-b \pm \sqrt{\text{discriminant}}}{2a} \)
- Simplify – Reduce fractions and simplify radicals
Mathematical concepts build on each other. If you are struggling with a topic, the real gap is usually one or two levels below where you think it is.
The Parabola
The graph of \( y = ax^2 + bx + c \) is a parabola:
- Opens up if \( a > 0 \)
- Opens down if \( a < 0 \)
- Vertex at \( x = -\frac{b}{2a} \)
- Axis of symmetry is the vertical line \( x = -\frac{b}{2a} \)
The x-intercepts (roots) are where the parabola crosses the x-axis.
Alternative Methods
Factoring
If \( ax^2 + bx + c \) factors nicely, this is often faster.
Example: \( x^2 – 5x + 6 = (x – 2)(x – 3) = 0 \), so \( x = 2 \) or \( x = 3 \)
Completing the Square
Useful for deriving the quadratic formula or converting to vertex form.
Graphing
Can find approximate solutions by identifying x-intercepts.
Vertex Form
Converting to vertex form reveals the vertex directly:
$$y = a(x – h)^2 + k$$
where \( (h, k) \) is the vertex.
Applications
Physics
- Projectile motion (height vs time)
- Falling objects
Engineering
- Optimization problems
- Area and volume calculations
Economics
- Profit maximization
- Supply and demand curves
Sum and Product of Roots
For \( ax^2 + bx + c = 0 \) with roots \( r_1 \) and \( r_2 \):
$$r_1 + r_2 = -\frac{b}{a}$$
$$r_1 \times r_2 = \frac{c}{a}$$
This can be useful for checking answers or constructing equations from roots.
How to verify quadratic roots
Substitute each reported root into \(ax^2+bx+c\). The result should be zero apart from rounding. Vieta’s formulas add a fast check: the roots should sum to \(-b/a\) and multiply to \(c/a\).
The discriminant \(b^2-4ac\) explains the root type. A positive value gives two distinct real roots, zero gives one repeated real root, and a negative value gives a complex-conjugate pair for real coefficients.
Keep the sign of b inside \(-b\) and use parentheses around the complete numerator. Dividing only the radical by \(2a\) is one of the most common handwritten errors.
For \(x^2-5x+6=0\), the discriminant is 1 and the formula gives \((5\pm1)/2\), so the roots are 3 and 2. Their sum is 5 and product is 6, matching the coefficients. The graph crosses the x-axis at both values.
When the discriminant is negative, write the square root as \(i\sqrt{|D|}\). Real-coefficient quadratics produce complex roots in conjugate pairs, so a root \(u+vi\) is accompanied by \(u-vi\).
Factoring can be faster for simple integer roots, but the quadratic formula works for every quadratic with \(a\ne0\). Completing the square gives the same result and explains where the formula comes from.
Common mistakes to avoid
- Entering a as zero, which makes the equation linear
- Dropping the plus-or-minus branch
- Calculating \((-b)^2\) instead of \(b^2\) inconsistently
- Rounding the discriminant before taking its square root
Useful algebra books and tools
For quadratic equations, use the discriminant to predict the root type and substitute the roots back into the original equation. The best way to make an algebra calculator useful is to pair its result with enough written practice to recognize the underlying pattern.
- Practice the method: Browse algebra textbooks and workbooks on Amazon. Choose a book with worked solutions and mixed problem sets, not only formula summaries.
- Check routine calculations: Browse scientific calculators on Amazon. Use one to verify arithmetic after you have written the algebraic steps yourself.
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What does the discriminant tell you about the solutions?
The discriminant (b² – 4ac) reveals the nature of solutions without solving. If positive, you get two different real roots. If zero, you get one repeated real root (the parabola touches the x-axis at one point). If negative, there are no real roots—only complex conjugate roots involving imaginary numbers.
When should I use the quadratic formula vs factoring?
Use factoring when the coefficients are small and you can quickly spot the factors. Use the quadratic formula when factoring isn’t obvious, coefficients are large or messy, or you need exact values including irrational roots. The quadratic formula always works; factoring is faster when it works but not always possible.
What are complex roots and when do they occur?
Complex roots occur when the discriminant is negative, so you’re taking the square root of a negative number. They come in conjugate pairs like 2 + 3i and 2 – 3i, where i = √(-1). Graphically, this means the parabola doesn’t cross the x-axis. Complex roots are real in advanced math and engineering applications.
How do I find the vertex of a parabola?
The x-coordinate of the vertex is x = -b/(2a). Plug this back into the equation to find the y-coordinate. Alternatively, convert to vertex form y = a(x-h)² + k, where (h,k) is the vertex. The vertex is either the maximum (if a 0) point of the parabola.
What is completing the square?
Completing the square rewrites ax² + bx + c by adding and subtracting a value to create a perfect square trinomial. For x² + bx, add (b/2)² to complete x² + bx + (b/2)² = (x + b/2)². This technique derives the quadratic formula and converts to vertex form y = a(x-h)² + k.
How are the sum and product of roots useful?
If roots are r₁ and r₂, then r₁ + r₂ = -b/a and r₁ × r₂ = c/a. Use these to: (1) check your solutions without re-solving, (2) construct a quadratic equation from known roots: x² – (sum)x + (product) = 0, (3) find information about roots without actually solving. These relationships come from factored form a(x-r₁)(x-r₂).
Why must a ≠ 0 in a quadratic equation?
If a = 0, the x² term disappears and you’re left with bx + c = 0, which is a linear equation (degree 1), not quadratic (degree 2). The quadratic formula would involve division by zero. Linear equations have at most one solution and are solved simply as x = -c/b.
What does the ± mean in the quadratic formula?
The ± (plus or minus) indicates there are two solutions: one using + and one using -. This gives x = (-b + √(b²-4ac))/(2a) and x = (-b – √(b²-4ac))/(2a). When the discriminant is zero, both formulas give the same answer, which is why there’s one repeated root. The ± captures the parabola’s symmetry about its vertex.
