Quadratic Equation Calculator
Use this quadratic equation calculator to solve equations in the standard form \(ax^2+bx+c=0\). Enter the coefficients \(a\), \(b\), and \(c\), and the calculator will find the roots, discriminant, vertex, axis of symmetry, y-intercept, and graph direction. The main quadratic formula is \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\] This formula works for real roots, repeated roots, and complex roots.
Solve a Quadratic Equation
Enter the coefficients for \(ax^2+bx+c=0\). The coefficient \(a\) cannot be zero because then the equation would not be quadratic.
Result
The discriminant \(D=b^2-4ac\) determines whether the equation has two real roots, one repeated real root, or two complex roots.
Quadratic Equation Formula
A quadratic equation is a second-degree polynomial equation in the form:
The quadratic formula solves any quadratic equation in standard form:
The expression inside the square root is called the discriminant:
The discriminant tells you the type of roots before you fully solve the equation.
When \(D>0\)
The quadratic equation has two distinct real roots. The parabola crosses the x-axis twice.
When \(D<0\)
The quadratic equation has two complex roots. The parabola does not cross the x-axis.
How to Use the Quadratic Equation Calculator
- Write your equation in standard form: \(ax^2+bx+c=0\).
- Enter the value of \(a\), the coefficient of \(x^2\).
- Enter the value of \(b\), the coefficient of \(x\).
- Enter the value of \(c\), the constant term.
- Choose the number of decimal places and answer format.
- Click Solve Quadratic Equation to see roots, discriminant, vertex, axis of symmetry, and graph information.
Quadratic Equation Formulas and Graph Features
| Feature | Formula | Meaning |
|---|---|---|
| Standard form | \[ax^2+bx+c=0\] | The usual form for applying the quadratic formula. |
| Quadratic formula | \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\] | Finds the roots of any quadratic equation. |
| Discriminant | \[D=b^2-4ac\] | Determines the number and type of roots. |
| Axis of symmetry | \[x=-\frac{b}{2a}\] | The vertical line through the vertex of the parabola. |
| Vertex x-coordinate | \[h=-\frac{b}{2a}\] | The x-coordinate of the turning point. |
| Vertex y-coordinate | \[k=f(h)=ah^2+bh+c\] | The minimum or maximum value of the quadratic function. |
| Vertex form | \[y=a(x-h)^2+k\] | Shows the vertex directly as \((h,k)\). |
| Y-intercept | \[(0,c)\] | The point where the graph crosses the y-axis. |
| Opens upward | \[a>0\] | The vertex is a minimum point. |
| Opens downward | \[a<0\] | The vertex is a maximum point. |
Worked Examples
Example 1: Solve a factorable quadratic equation
Solve \(x^2-5x+6=0\).
The roots are \(x=2\) and \(x=3\). This equation can also be factored as \((x-2)(x-3)=0\).
Example 2: Solve a quadratic with one repeated root
Solve \(x^2-6x+9=0\).
The equation has one repeated real root: \(x=3\). The graph touches the x-axis at the vertex.
Example 3: Solve a quadratic with complex roots
Solve \(x^2+4x+8=0\).
The roots are \(x=-2+2i\) and \(x=-2-2i\). Since the discriminant is negative, there are no real x-intercepts.
Example 4: Find the vertex
Find the vertex of \(y=x^2-4x+1\).
The vertex is \((2,-3)\). Since \(a=1>0\), the parabola opens upward and the vertex is a minimum point.
Example 5: Use the discriminant only
Determine the type of roots for \(2x^2+3x+7=0\).
Since \(D<0\), the equation has two complex conjugate roots and no real roots.
Complete Guide to Quadratic Equations
A quadratic equation is an equation where the highest power of the variable is \(2\). The standard form is \(ax^2+bx+c=0\), where \(a\), \(b\), and \(c\) are constants and \(a\ne0\). The condition \(a\ne0\) is essential because if \(a=0\), the term \(ax^2\) disappears and the equation becomes linear rather than quadratic.
Quadratic equations appear throughout algebra, geometry, physics, finance, engineering, and data modeling. They are used to model projectile motion, area problems, profit functions, parabolic graphs, optimization, and situations where a quantity depends on the square of another quantity. The graph of a quadratic function is a parabola. Depending on the value of \(a\), the parabola opens upward or downward.
Parts of a quadratic equation
In \(ax^2+bx+c=0\), the coefficient \(a\) controls the direction and width of the parabola. If \(a>0\), the parabola opens upward. If \(a<0\), the parabola opens downward. A larger absolute value of \(a\) makes the parabola narrower, while a smaller absolute value makes it wider. The coefficient \(b\) helps determine the location of the vertex and axis of symmetry. The constant \(c\) is the y-intercept because when \(x=0\), the function value is \(c\).
What are roots?
The roots of a quadratic equation are the values of \(x\) that make the equation equal to zero. They are also called solutions, zeros, or x-intercepts. If a quadratic function is written as \(y=ax^2+bx+c\), the roots occur where \(y=0\). On the graph, real roots are the points where the parabola crosses or touches the x-axis.
A quadratic equation can have two distinct real roots, one repeated real root, or two complex roots. The discriminant tells which situation occurs. If the parabola crosses the x-axis twice, there are two real roots. If it touches the x-axis once at the vertex, there is one repeated real root. If it does not touch the x-axis, the roots are complex.
The quadratic formula
The quadratic formula is the most reliable method for solving quadratic equations because it works for all quadratics in standard form. The formula is:
The symbol \(\pm\) means “plus or minus.” It gives two possible solutions:
If the discriminant is zero, both formulas give the same value, so the equation has one repeated root. If the discriminant is negative, the square root involves \(\sqrt{-1}\), so the roots are complex.
The discriminant
The discriminant is the expression \(b^2-4ac\). It is often written as \(D\) or \(\Delta\). The discriminant is important because it tells the type of roots without requiring the full solution.
If \(D>0\), there are two distinct real roots. If \(D=0\), there is one repeated real root. If \(D<0\), there are two complex conjugate roots. The discriminant is a quick diagnostic tool for understanding the equation before solving completely.
Solving by factoring
Factoring is another method for solving quadratic equations. It works well when the quadratic can be rewritten as a product of two simpler expressions. For example:
Using the zero product property, if \((x-2)(x-3)=0\), then \(x-2=0\) or \(x-3=0\). Therefore, \(x=2\) or \(x=3\). Factoring is fast when it works, but not every quadratic factors easily over the integers. The quadratic formula always works.
Solving by completing the square
Completing the square transforms a quadratic into a perfect-square expression. It is the method used to derive the quadratic formula. For a simple example:
So the roots are \(x=-1\) and \(x=-5\). Completing the square is also useful for rewriting a quadratic in vertex form.
Vertex of a quadratic
The vertex is the turning point of the parabola. If the parabola opens upward, the vertex is the minimum point. If the parabola opens downward, the vertex is the maximum point. For \(y=ax^2+bx+c\), the x-coordinate of the vertex is:
The y-coordinate is found by substituting \(h\) into the function:
The vertex is \((h,k)\). The calculator finds this automatically because the vertex gives important graph information.
Axis of symmetry
The axis of symmetry is the vertical line that passes through the vertex and divides the parabola into two mirror-image halves. Its formula is:
This is the same as the x-coordinate of the vertex. If a quadratic has two real roots, the axis of symmetry lies exactly halfway between them. For example, if the roots are \(2\) and \(6\), the axis of symmetry is \(x=4\).
Quadratic graph direction
The sign of \(a\) determines the graph direction. If \(a>0\), the parabola opens upward and the vertex is a minimum. If \(a<0\), the parabola opens downward and the vertex is a maximum. This graph direction is useful for optimization problems. For example, a profit function that opens downward has a maximum profit at the vertex.
Y-intercept
The y-intercept of \(y=ax^2+bx+c\) is \((0,c)\). This happens because when \(x=0\), the terms \(ax^2\) and \(bx\) become zero, leaving \(y=c\). The y-intercept is helpful when sketching a parabola.
Real roots and complex roots
Real roots can be plotted on the number line and appear as x-intercepts on the graph. Complex roots include the imaginary unit \(i\), where \(i^2=-1\). When a quadratic has real coefficients and complex roots, the roots always appear as a conjugate pair. For example, if one root is \(2+3i\), the other root is \(2-3i\).
Complex roots do not show as x-intercepts on the real coordinate plane, but they are still valid solutions of the equation. The quadratic formula naturally produces them when the discriminant is negative.
Quadratic equations in physics
Quadratic equations appear often in physics, especially in projectile motion. A height function might look like \(h(t)=-16t^2+vt+s\) in feet or \(h(t)=-4.9t^2+vt+s\) in meters. The roots can represent the times when the object hits the ground. The vertex can represent the maximum height. This is one reason quadratics are important beyond pure algebra.
Quadratic equations in area problems
Quadratic equations also appear in geometry and area problems. If the length and width of a rectangle depend on a variable, multiplying them often creates an \(x^2\) term. For example, if a rectangle has sides \(x+3\) and \(x+5\), its area is \((x+3)(x+5)=x^2+8x+15\). Solving area equations may require factoring or the quadratic formula.
Quadratic equations in optimization
Quadratic functions are used in optimization because their graphs have clear maximum or minimum points. If the parabola opens upward, the vertex gives the minimum value. If the parabola opens downward, the vertex gives the maximum value. Businesses may use quadratic models for revenue or profit, while physics problems may use them for maximum height or minimum distance.
Choosing a solving method
There are several methods for solving quadratic equations: factoring, completing the square, graphing, and using the quadratic formula. Factoring is fastest when the quadratic factors neatly. Completing the square is useful for deriving formulas and converting to vertex form. Graphing gives a visual estimate of roots and shape. The quadratic formula is the most universal because it works every time.
In exams, students are often expected to recognize when factoring is easy. However, when coefficients are large, decimals are involved, or roots are complex, the quadratic formula is usually the safest method.
Common mistakes with quadratic equations
A common mistake is forgetting that the entire numerator of the quadratic formula is divided by \(2a\). The formula is \(\frac{-b\pm\sqrt{b^2-4ac}}{2a}\), not \(-b\pm\frac{\sqrt{b^2-4ac}}{2a}\). Another common mistake is losing the negative sign when substituting \(b\). If \(b=-5\), then \(-b=5\).
Students also often make sign errors in the discriminant. The expression is \(b^2-4ac\). If \(c\) is negative, then \(-4ac\) may become positive. Careful substitution with parentheses helps avoid mistakes.
How to check your solutions
To check a root, substitute it back into the original equation. If the equation becomes true, the root is correct. For example, for \(x^2-5x+6=0\), test \(x=2\):
Since the expression equals zero, \(x=2\) is a solution. Checking is especially useful when answers involve radicals, fractions, or complex numbers.
Summary
A quadratic equation has the form \(ax^2+bx+c=0\) with \(a\ne0\). The quadratic formula \(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\) solves every quadratic equation. The discriminant \(D=b^2-4ac\) tells whether the equation has two real roots, one repeated real root, or two complex roots. The graph is a parabola, and its vertex is found using \(x=-\frac{b}{2a}\). Together, these formulas give a complete understanding of the equation, its solutions, and its graph.
Common Mistakes with Quadratic Equations
Forgetting that \(a\ne0\)
If \(a=0\), the equation is not quadratic. It becomes a linear equation.
Dropping the plus-minus sign
The quadratic formula uses \(\pm\), so it usually gives two roots.
Making sign errors with \(-b\)
If \(b\) is negative, then \(-b\) is positive. Use parentheses when substituting values.
Misreading the discriminant
If \(D<0\), the roots are complex, not “no solution.” There are no real roots, but there are complex roots.
Related Calculators and Study Tools
These related tools can help with algebra, functions, graphing, and equation solving:
Quadratic Equation Calculator FAQs
What is a quadratic equation?
A quadratic equation is an equation in the form \(ax^2+bx+c=0\), where \(a\ne0\).
What is the quadratic formula?
The quadratic formula is \[x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\]
What is the discriminant?
The discriminant is \(D=b^2-4ac\). It tells whether the quadratic equation has two real roots, one repeated real root, or two complex roots.
How many solutions can a quadratic equation have?
A quadratic equation can have two distinct real solutions, one repeated real solution, or two complex solutions.
When does a quadratic have two real roots?
A quadratic has two distinct real roots when the discriminant is positive: \(D>0\).
When does a quadratic have one repeated root?
A quadratic has one repeated real root when the discriminant is zero: \(D=0\).
When does a quadratic have complex roots?
A quadratic has complex roots when the discriminant is negative: \(D<0\).
What is the vertex of a quadratic?
The vertex is the turning point of the parabola. Its x-coordinate is \(x=-\frac{b}{2a}\).
What is the axis of symmetry?
The axis of symmetry is the vertical line through the vertex. For \(y=ax^2+bx+c\), it is \(x=-\frac{b}{2a}\).
Can every quadratic equation be solved by the quadratic formula?
Yes. The quadratic formula works for every quadratic equation in standard form, including equations with real, repeated, or complex roots.
