Quadratic Equation Calculator | Solve Ax² + Bx + C Instantly

How to use the quadratic equation calculator

1. Select a mode. Solve Quadratic Equation finds the roots; the other modes focus on features such as the vertex, discriminant, transformations, or sign intervals.
2. Enter \(a\), \(b\), and \(c\). The value of \(a\) cannot be zero because an equation with \(a=0\) is linear.
3. Select Calculate. Read the result together with the displayed method or interpretation.
4. Check the answer by substituting each root into the original equation. A correct root makes the left side equal to zero, apart from small rounding differences for decimal answers.

The expression \(b^2-4ac\) is the discriminant. It predicts the kind of roots before the complete calculation is carried out.

Worked example: two distinct real roots

Solve \(2x^2-5x-3=0\). The coefficients are \(a=2\), \(b=-5\), and \(c=-3\). First calculate the discriminant:

Because \(\Delta=49>0\), the equation has two distinct real roots. Substitute the coefficients into the quadratic formula:

Using the plus sign gives \(x=3\). Using the minus sign gives \(x=-\frac12\). A quick check confirms both answers: \(2(3)^2-5(3)-3=0\), and \(2(-\frac12)^2-5(-\frac12)-3=0\).

Worked example: one repeated root

Solve \(x^2-6x+9=0\). Here \(a=1\), \(b=-6\), and \(c=9\).

A zero discriminant means the two formula branches meet at one repeated root:

The same result appears by factoring: \(x^2-6x+9=(x-3)^2\). On the graph, the parabola touches the \(x\)-axis at \((3,0)\) and turns around instead of crossing it.

Choose a solving method that matches the equation

Common input mistakes and how to avoid them

• Forgetting to rearrange: the calculator expects every term on one side and zero on the other.
• Losing a negative sign: in \(x^2-4x-5=0\), the correct entries are \(1,-4,-5\).
• Entering a zero leading coefficient: \(0x^2+bx+c=0\) is not quadratic.
• Using rounded coefficients too early: keep exact fractions or sufficient decimal places until the final answer.
• Reporting only one root: when \(\Delta>0\), both the plus and minus branches must be evaluated.
• Confusing a root with the vertex: roots have \(y=0\); the vertex is the turning point and does not normally lie on the \(x\)-axis.

What is a Quadratic Equation? What is a Quadratic?

A quadratic equation is a second-degree polynomial equation in a single variable \(x\). This means the highest power of the variable \(x\) is 2.

The standard form of a quadratic equation is:

Where:

• \(x\) is the variable.
• \(a\), \(b\), and \(c\) are coefficients, which are known numbers.
• Crucially, \(a\) cannot be equal to 0 (if \(a=0\), the equation becomes linear, not quadratic).

The term "quadratic" itself comes from the Latin word "quadratus," meaning square, referring to the \(x^{2}\) term. "What is a quadratic" generally refers to either a quadratic equation or a quadratic function/expression.

What is the Quadratic Formula? What's the Quadratic Formula?

The quadratic formula is a formula that provides the solution(s) to any quadratic equation in the standard form \(ax^{2} + bx + c = 0\).

The formula is:

This formula gives two possible values for \(x\) (due to the ± symbol), which are the roots or solutions of the quadratic equation.

What is a, b, and c in the Quadratic Formula? What is 'a' in a Quadratic Equation/Function?

In the standard form of a quadratic equation, \(ax^{2} + bx + c = 0\), and consequently in the quadratic formula:

• \(a\) is the coefficient of the \(x^{2}\) term (the quadratic term).
• \(b\) is the coefficient of the \(x\) term (the linear term).
• \(c\) is the constant term (the term without \(x\)).

It's essential to correctly identify \(a\), \(b\), and \(c\) from a given quadratic equation before plugging them into the formula. Remember that \(a ≠ 0\).

How to Solve Quadratic Equations? How to Do Quadratics/Quadratic Equations?

There are several methods to solve quadratic equations (i.e., find the values of \(x\) that satisfy the equation):

1. Factoring:
   • Rewrite the quadratic equation in standard form (\(ax^{2} + bx + c = 0\)).
   • Factor the quadratic expression on the left side into two linear factors.
   • Set each linear factor equal to zero and solve for \(x\). This method is often quickest if the quadratic expression is easily factorable.
2. Using the Quadratic Formula:
   • This method works for all quadratic equations.
   • Identify \(a\), \(b\), and \(c\) from the standard form.
   • Substitute these values into the formula: \(x = [-b ± \sqrt{b^{2} - 4ac}] / 2a\) and calculate the value(s) of \(x\).
3. Completing the Square:
   • This method involves manipulating the equation algebraically to create a perfect square trinomial on one side, which can then be solved by taking the square root of both sides.
   • It's a fundamental method that also leads to the derivation of the quadratic formula itself.
4. Graphing:
   • Graph the corresponding quadratic function \(y = ax^{2} + bx + c\).
   • The \(x\)-intercepts (where the graph crosses the \(x\)-axis, i.e., where \(y=0\)) are the real solutions to the quadratic equation \(ax^{2} + bx + c = 0\). This method is good for visualization and finding approximate solutions but may not be precise for exact values.

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How to Use the Quadratic Formula? How to Do/Calculate Quadratic Formula?

To use the quadratic formula \(x = [-b ± \sqrt{b^{2} - 4ac}] / 2a\):

1. Write the equation in standard form: Ensure your quadratic equation is arranged as \(ax^{2} + bx + c = 0\).
2. Identify coefficients: Determine the values of \(a\) (coefficient of \(x^{2}\)), \(b\) (coefficient of \(x\)), and \(c\) (the constant term). Pay attention to signs.
3. Substitute into the formula: Carefully plug the values of \(a\), \(b\), and \(c\) into the quadratic formula.
4. Calculate the discriminant: First, calculate the value inside the square root: \(D = b^{2} - 4ac\). This is called the discriminant.
5. Calculate the square root: Find the square root of the discriminant, \(√D\).
   • If \(D > 0\), there are two distinct real solutions.
   • If \(D = 0\), there is one real solution (a repeated root).
   • If \(D \lt 0\), there are two complex conjugate solutions (no real solutions).
6. Solve for x: Calculate the two possible values for \(x\) using the ± sign:
   • \(x_{1} = (-b + √D) / 2a\)
   • \(x_{2} = (-b - √D) / 2a\)
7. Simplify: Simplify your answers if possible.

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When to Use the Quadratic Formula? What is the Quadratic Formula Used For? What Does it Do?

The quadratic formula is used to find the roots (or solutions) of any quadratic equation of the form \(ax^{2} + bx + c = 0\).

When to use it:

• When factoring the quadratic expression is difficult or not immediately obvious.
• When you need to find exact solutions, including irrational or complex roots, which factoring might not easily reveal.
• It is a universal method that always works, unlike factoring, which is only straightforward for certain types of quadratics.
• It's also used in the derivation of other mathematical concepts and in various applications in physics, engineering, and economics where quadratic relationships arise.

"What does the quadratic formula do?" It provides a direct method to calculate the values of \(x\) that make the quadratic equation true.

How to Factor Quadratic Equations? How to Factor Quadratics?

Factoring a quadratic equation \(ax^{2} + bx + c = 0\) involves rewriting the quadratic expression as a product of two linear factors.

Steps (for \(a=1\), i.e., \(x^{2} + bx + c\)):

1. Find two numbers that multiply to give \(c\) and add to give \(b\). Let these numbers be \(p\) and \(q\).
2. The factored form will be \((x + p)(x + q) = 0\).
3. Set each factor to zero to find the solutions: \(x + p = 0\) => \(x = -p\), and \(x + q = 0\) => \(x = -q\).

Steps (for \(a ≠ 1\), i.e., \(ax^{2} + bx + c\) - often called the "ac method" or grouping):

1. Find two numbers that multiply to give \(a × c\) and add to give \(b\). Let these be \(p\) and \(q\).
2. Rewrite the middle term \(bx\) as \(px + qx\): \(ax^{2} + px + qx + c = 0\).
3. Factor by grouping: Group the first two terms and the last two terms, and factor out the greatest common factor (GCF) from each group. \(x(ax + p) + y(ax + p) = 0\) (where y is the GCF of qx and c)
4. Factor out the common binomial factor: \((ax + p)(x + y) = 0\).
5. Set each factor to zero and solve.

What is a Quadratic Function? How to Graph Quadratic Functions?

A quadratic function has a nonzero squared-term coefficient and can be written in standard form or vertex form:

Its graph is a parabola. When \(a\gt 0\), the parabola opens upward and its vertex is a minimum. When \(a\lt 0\), it opens downward and its vertex is a maximum. A larger value of \(|a|\) makes the graph narrower, while a value of \(|a|\) closer to zero makes it wider.

To sketch the graph accurately, find the vertex, axis of symmetry, intercepts, and one or two extra points. Plot corresponding points at equal horizontal distances from the axis of symmetry because a parabola is symmetric.

• Axis of symmetry: \(x=-\frac{b}{2a}\).
• Vertex: calculate \(h=-\frac{b}{2a}\), then calculate \(k=f(h)\). The vertex is \((h,k)\).
• Y-intercept: set \(x=0\). Because \(f(0)=c\), the point is \((0,c)\).
• X-intercepts: solve \(ax^2+bx+c=0\). Depending on the discriminant, there may be two, one, or no real x-intercepts.

How to Find the Vertex of a Quadratic Function?

For \(f(x)=ax^2+bx+c\), first find the horizontal coordinate of the vertex and then substitute it into the function:

The vertex is \((h,k)\). If the function is already written as \(f(x)=a(x-h)^2+k\), the vertex can be read directly. Pay attention to the sign inside the bracket: \((x+3)^2\) is the same as \((x-(-3))^2\), so its horizontal coordinate is \(-3\).

The vertex is the minimum point when \(a\gt0\) and the maximum point when \(a\lt0\). It also lies on the axis of symmetry, so it provides a useful check when graphing.

What is the Discriminant of a Quadratic Equation?

The discriminant is the expression under the square root in the quadratic formula:

• If \(\Delta\gt0\), there are two distinct real roots, and the parabola crosses the x-axis twice.
• If \(\Delta=0\), there is one repeated real root, and the vertex touches the x-axis.
• If \(\Delta\lt0\), there are two complex conjugate roots and no real x-intercepts.

The discriminant describes the type of solution; it does not replace the remaining steps needed to calculate the roots. For example, \(\Delta=49\) tells you that two real roots exist and that the square-root step will simplify neatly, but you must still substitute into the full formula.

What are the Roots of a Quadratic Equation? How to Find the Roots?

The roots of a quadratic equation \(ax^{2} + bx + c = 0\) are the values of \(x\) that make the equation true. They are also known as the solutions or zeros of the equation (or the x-intercepts of the corresponding quadratic function's graph).

How to find the roots:

• Use the quadratic formula: \(x = [-b ± \sqrt{b^{2} - 4ac}] / 2a\).
• Factor the quadratic expression and set each factor to zero.
• Complete the square.

The discriminant (\(b^{2} - 4ac\)) determines how many and what type of roots the equation has (two distinct real, one repeated real, or two complex conjugate roots).

How to Find X-intercepts / Y-intercept of a Quadratic Function?

• X-intercepts: These are the points where the graph of \(f(x) = ax^{2} + bx + c\) crosses the \(x\)-axis. At these points, \(f(x) = 0\). So, to find the x-intercepts, you solve the quadratic equation \(ax^{2} + bx + c = 0\). The solutions are the x-coordinates of the x-intercepts. (See "How to Find the Roots").
• Y-intercept: This is the point where the graph crosses the \(y\)-axis. At this point, \(x = 0\). To find the y-intercept, substitute \(x = 0\) into the function: \(f(0) = a(0)^{2} + b(0) + c = c\). So, the y-intercept is the point \((0, c)\).

How to Find the Range of a Quadratic Function?

The range of a function is the set of all possible output values (\(f(x)\) or \(y\) values).

For a quadratic function \(f(x) = ax^{2} + bx + c\), the range depends on the direction the parabola opens and the \(y\)-coordinate of its vertex (\(k\)).

• If \(a > 0\) (parabola opens upwards), the vertex \((h, k)\) is the minimum point. The range is \(f(x) ≥ k\), or in interval notation, \([k, ∞)\).
• If \(a \lt 0\) (parabola opens downwards), the vertex \((h, k)\) is the maximum point. The range is \(f(x) ≤ k\), or in interval notation, \((-∞, k]\).

To find the range, first find the vertex \((h, k)\). The \(y\)-coordinate \(k\) is the key value.

How to Write a Quadratic Function in Standard Form?

The standard form of a quadratic function is \(f(x) = ax^{2} + bx + c\).

If you have a quadratic function in a different form (e.g., vertex form \(f(x) = a(x - h)^{2} + k\), or factored form \(f(x) = a(x - r_{1})(x - r_{2})\)), you can convert it to standard form by expanding and simplifying:

• From Vertex Form \(f(x) = a(x - h)^{2} + k\):
  1. Expand \((x - h)^{2}\) to get \(x^{2} - 2hx + h^{2}\).
  2. Distribute the \(a\): \(a(x^{2} - 2hx + h^{2}) = ax^{2} - 2ahx + ah^{2}\).
  3. Add \(k\): \(f(x) = ax^{2} - 2ahx + ah^{2} + k\).
  4. Combine constants: Here, \(b = -2ah\) and \(c = ah^{2} + k\).
• From Factored Form \(f(x) = a(x - r_{1})(x - r_{2})\):
  1. Expand \((x - r_{1})(x - r_{2})\) using FOIL (First, Outer, Inner, Last) to get \(x^{2} - r_{2}x - r_{1}x + r_{1}r_{2}\), which simplifies to \(x^{2} - (r_{1} + r_{2})x + r_{1}r_{2}\).
  2. Distribute \(a\) across the three terms:
     \[f(x)=a(x^2-(r_1+r_2)x+r_1r_2)\]

     Then simplify each product:

     \[f(x)=ax^2-a(r_1+r_2)x+ar_1r_2\]

How to Find a Quadratic Equation from a Graph?

To find the equation of a quadratic function from its graph, you need to identify key features:

1. Identify the Vertex \((h, k)\): If the vertex is clearly visible, you can use the vertex form \(f(x) = a(x - h)^{2} + k\).
2. Identify another point \((x, y)\) on the parabola: Substitute the coordinates of the vertex (\(h, k\)) and the other point (\(x, y\)) into the vertex form and solve for \(a\).
3. Write the equation: Once you have \(a\), \(h\), and \(k\), write the equation in vertex form. You can then expand it to standard form if needed.

Alternatively, if the x-intercepts (roots) \(r_{1}\) and \(r_{2}\) are clear:

1. Use the factored form: \(f(x) = a(x - r_{1})(x - r_{2})\).
2. Identify another point \((x, y)\) on the parabola (often the y-intercept or vertex).
3. Substitute the roots and the other point into the factored form and solve for \(a\).
4. Write the equation in factored form. Expand to standard form if needed.

If you have three distinct points, you can set up a system of three linear equations using the standard form \(y = ax^{2} + bx + c\) and solve for \(a\), \(b\), and \(c\), but this is more algebraically intensive.

How to Solve Quadratic Inequalities? What is the Solution Set?

A quadratic inequality involves expressions like \(ax^{2} + bx + c > 0\), \(ax^{2} + bx + c \lt 0\), \(ax^{2} + bx + c ≥ 0\), or \(ax^{2} + bx + c ≤ 0\).

Steps to solve:

1. Rearrange: Get the inequality into a form where one side is 0 (e.g., \(ax^{2} + bx + c > 0\)).
2. Find Critical Points (Roots): Solve the corresponding quadratic equation \(ax^{2} + bx + c = 0\) to find its roots (the x-intercepts of the parabola). These roots divide the number line into intervals.
3. Test Intervals:
   • Choose a test value from within each interval created by the roots.
   • Substitute this test value into the original inequality (or the rearranged quadratic expression \(ax^{2} + bx + c\)).
   • Determine if the inequality is true or false for that interval.
   Alternatively, sketch the graph of the parabola. The intervals where the parabola is above the x-axis satisfy \(f(x) > 0\), and where it's below satisfy \(f(x) \lt 0\).
4. Write the Solution Set: Express the solution as an interval or a union of intervals, paying attention to whether the inequality is strict (\(\lt , >\)) or inclusive (\(≤, ≥\)). For inclusive inequalities, the roots themselves are part of the solution.

"What is the solution set of the quadratic inequality" refers to this interval or union of intervals.

How to check calculator answers

Substitution is the most reliable check. If the calculator returns \(r\), evaluate \(ar^2+br+c\). An exact root produces zero. A rounded decimal root may produce a very small value such as \(0.0001\); that difference is caused by rounding rather than a different solution. You can also compare the sum and product of the roots with Vieta's relationships:

For a graph check, real roots should match the \(x\)-intercepts, the axis of symmetry should pass through the vertex, and the graph should open upward when \(a>0\) or downward when \(a\lt 0\).

Continue learning

Use the quadratic equation study guide for a lesson-style explanation, review solving quadratics for GCSE algebra, or practise broader equation skills with the algebra calculator with steps.

Reference and scope

The formulas and discriminant rules on this page agree with the treatment in OpenStax College Algebra 2e. Calculator results are mathematical computations, not official exam-board marking guidance. Decimal displays may be rounded, so use exact forms when a question requires an exact answer.