Roots
- $x_1$: —
- $x_2$: —
Step-by-step
See the key steps used to reach the roots:
Quadratic Equation Solver
This tool solves quadratic equations in the standard form $ax^2 + bx + c = 0$. Enter $a$, $b$, and $c$ to compute the discriminant ($\Delta$) and the roots using the quadratic formula, then interpret whether the results are two real roots, one repeated root, or a complex conjugate pair. For quadratic roots, discriminants, and related algebra methods, explore Algebra.
Solve equations in the form:
$$ax^2 + bx + c = 0$$
If a = 0, the equation becomes linear (bx + c = 0).
What Is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree two. It involves a squared variable term and can produce up to two solutions, known as roots. These solutions describe where the equation intersects the horizontal axis.
Quadratic equations appear in physics, engineering, finance, and many applied sciences. Understanding how their solutions behave is essential for modeling curved relationships.
Standard Form and Coefficients
The standard form of a quadratic equation is written as $ax^2 + bx + c = 0$. The values $a$, $b$, and $c$ are called coefficients and determine the shape and position of the curve.
The Quadratic Formula
The quadratic formula provides a direct method to compute the roots of any quadratic equation. It works by substituting the coefficients into a single expression.
$$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$$
The expression under the square root is called the discriminant. Its value determines whether the equation has two real solutions, one repeated solution, or complex roots.
- $\Delta > 0$: two distinct real roots
- $\Delta = 0$: one repeated real root
- $\Delta < 0$: two complex roots (a conjugate pair)
How to Solve a Quadratic Equation
Solving a quadratic equation follows a consistent sequence of steps. These steps ensure the formula is applied correctly and the results are interpreted properly.
First, identify the coefficients $a$, $b$, and $c$ from the equation. Next, substitute them into the quadratic formula. Then evaluate the discriminant and simplify the result. Finally, interpret the roots based on the discriminant value.
Worked Examples
Example 1: Two Real Roots
Use this when the discriminant is positive, so the equation has two distinct real solutions. We'll plug the coefficients into the quadratic formula and simplify.
Solve: $$x^2 - 5x + 6 = 0$$ Coefficients: $$a=1,\; b=-5,\; c=6$$
Compute the discriminant: $$\Delta = b^2 - 4ac = (-5)^2 - 4(1)(6) = 25 - 24 = 1$$ $$\Delta > 0$$ There are two distinct real roots.
Apply the quadratic formula: $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-5) \pm \sqrt{1}}{2(1)} = \frac{5 \pm 1}{2}$$ So the two solutions are: $$x_1 = \frac{5+1}{2} = 3,\qquad x_2 = \frac{5-1}{2} = 2$$
Example 2: Repeated Root
Use this when the discriminant equals zero, so both solutions collapse into one repeated real root. The quadratic formula still applies, but the $\pm$ part becomes the same value.
Solve: $$x^2 - 4x + 4 = 0$$ Coefficients: $$a=1,\; b=-4,\; c=4$$
Compute the discriminant: $$\Delta = b^2 - 4ac = (-4)^2 - 4(1)(4) = 16 - 16 = 0$$ $$\Delta = 0$$ There is one repeated real root.
Apply the quadratic formula: $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-(-4) \pm \sqrt{0}}{2(1)} = \frac{4 \pm 0}{2}$$ So the solution is: $$x = 2$$
Example 3: No Real Solutions
Use this when the discriminant is negative, so the square root introduces an imaginary value. The results are a complex conjugate pair.
Solve: $$x^2 + 2x + 5 = 0$$ Coefficients: $$a=1,\; b=2,\; c=5$$
Compute the discriminant: $$\Delta = b^2 - 4ac = (2)^2 - 4(1)(5) = 4 - 20 = -16$$ $$\Delta < 0$$ There are no real roots, and the solutions are complex.
Apply the quadratic formula: $$x = \frac{-b \pm \sqrt{\Delta}}{2a} = \frac{-2 \pm \sqrt{-16}}{2(1)}$$ Rewrite the square root: $$\sqrt{-16} = 4i$$ So the solutions are: $$x = \frac{-2 \pm 4i}{2} = -1 \pm 2i$$
If you need to simplify coefficients before solving, the GCD Calculator can help reduce common factors. For quick checks on integer values used in examples, you may also use the Prime Number Checker.
Keep exploring in Mathematics or jump back to Calculators to browse more tools.
Questions About the Quadratic Solver
Quick answers about quadratic equations, roots, and the quadratic formula.
A quadratic equation is an equation that can be written in the form $ax^2 + bx + c = 0$, where $a$ is not zero.
The discriminant $b^2 - 4ac$ indicates the type of solutions: two real roots, one repeated real root, or complex roots.
If $a = 0$, the equation is not quadratic. It becomes linear ($bx + c = 0$) if $b$ is not zero.
Yes. If the discriminant is negative, the solutions are complex and there are no real roots.
No. Roots can be integers, fractions, irrational numbers, or complex numbers depending on the coefficients.
For readability, results may be rounded to a limited number of decimal places. If the exact value is irrational, the decimal output is an approximation.
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