Quadratic simultaneous equations solver : With Step by step Solution/Explanation

A free online quadratic simultaneous equation solver that solve any system of equations, Shows all the workings step by step. Before we dive into how the online solver works, let’s get some basic on what qualifies as quadratic simultaneous equation. In particular, we will consider a system of quadratic and linear equation

A linear equation is an equation of the form y=ax+b, with the power of the variable x been one. On the other hand a quadratic equation is an equation of the form ax^2+bx+c where a\neq 0 . A combination of a linear and a quadratic forms a perfect system of equations or a pair of simultaneous equation. It is possible to solve such a system through the substitution method.

Examples of Quadratic simultaneous equation

1. 3x^2+4x-2=0,x+y=9
2. x^2+x=2,x-y=7
3. 2x^2+3x-12=0,x+3y=4
4. 6x^2+4x=12,x+2y=6
5. x^2+x=0,3x+y=2
6. x^2+8y=13;2y+x=2
7. (x^2)+(y^2)=1,y=2x-5

Free Quadratic Simultaneous Equation Solver with Steps

Solving systems of linear and quadratic equations requires more than just the final answer. Our online calculator provides detailed, step-by-step solutions so you can understand the process clearly. Whether you’re working with linear equations, quadratic equations, or a mix of both, this solver delivers accurate results quickly. You can even solve systems with multiple variables and up to 5 equations.

For better visualization, EquationCalc’s solver also includes a graphing tool that shows the graphical solution of simultaneous equations.

When solving a system involving a linear and a quadratic equation, there are usually two solution pairs. Quadratic equations represent conic sections such as circles, parabolas, ellipses, or hyperbolas, while linear equations represent straight lines. The solutions are the intersection points of these graphs.

Example 1: Quadratic Simultaneous Equation

1. y = x + 2
2. y = x^2 + 3x

Substitute y from equation (1) into equation (2):

x + 2 = x^2 + 3x

Rearranging gives: x^2 + 2x - 2 = 0

This quadratic can be solved using factorization:

x^2 + 2x - 2 = 0

(x - 2)(x + 1) = 0

So, x = 2 or x = -1

Substitute these values into equation (1) to find y:

y = 2 + 2 = 4

y = -1 + 2 = 1

Thus, the solutions are:

(2, 4) and (-1, 1)

Graphically, these represent the intersection points of the line y = x + 2 and the parabola y = x^2 + 3x.

View more solved examples: Algebra Examples with Steps

Try our solver for linear systems: Systems of Equations Solver

Simultaneous Linear Equations Solver

Our free online solver works with linear and non-linear equations of up to 5 variables. The premium package provides step-by-step explanations, making it ideal for learning and assignments. Solve for x and y instantly using EquationCalc’s online calculators.

Two-Variable Linear Equations Calculator

To solve a system with two variables, you need at least two equations. Similarly, three variables require three equations. If the number of variables exceeds the number of equations, the system has no solution. A system may also lack solutions if the graphs do not intersect.

Step-by-Step Simultaneous Equation Solver

EquationCalc’s solver is unique because it shows every step of the solution process. This makes it easier to understand how answers are derived.

How to Use EquationCalc’s Solver

How it Works

Here’s a worked example to demonstrate:

Step 1: Enter your equations separated by commas (,) or semicolons (;). Without separators, the calculator will return an error.

Step 2: Click “Calculate Now” to get instant solutions.

Step 3: Use the solutions for assignments or math problems. You can copy or print them for later use.

Learn Quadratic Simultaneous Equations with Examples

Learning through examples is the best way to master algebra. Explore our solved examples, each with step-by-step explanations, to guide you in using the calculator effectively.

See more: Quadratic Simultaneous Equations Examples

Accepted Math Symbols

• + Addition
• - Subtraction
• * Multiplication
• / Division
• ^ Exponentiation
• sqrt Square root

Pi: Represents the constant \pi

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