Solve The Following Quadratic Linear System of Equations Calculator
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Reviewed by Calculator Editorial Team
A quadratic linear system of equations consists of one quadratic equation and one linear equation. This calculator solves such systems by substitution or elimination methods.
Introduction
Quadratic linear systems involve solving two equations where one is quadratic (degree 2) and the other is linear (degree 1). These systems can have zero, one, or two solutions depending on the equations.
The general form of a quadratic linear system is:
1. \( ax^2 + bx + c = 0 \)
2. \( dx + e = 0 \)
To solve this system, we typically solve the linear equation for one variable and substitute into the quadratic equation.
How to Use the Calculator
Enter the coefficients for both equations in the calculator panel on the right. The calculator will:
- Solve the linear equation for one variable
- Substitute into the quadratic equation
- Solve the resulting quadratic equation
- Display all solutions if they exist
The calculator handles all cases including no solution, one solution, and two solutions.
Worked Example
Let's solve the system:
1. \( x^2 - 5x + 6 = 0 \)
2. \( 2x - 4 = 0 \)
Step 1: Solve the linear equation for x:
\( 2x - 4 = 0 \)
\( 2x = 4 \)
\( x = 2 \)
Step 2: Substitute x = 2 into the quadratic equation:
\( (2)^2 - 5(2) + 6 = 0 \)
\( 4 - 10 + 6 = 0 \)
\( 0 = 0 \)
This shows that x = 2 is a solution to both equations. To find other solutions, we can factor the quadratic equation:
\( x^2 - 5x + 6 = (x-2)(x-3) = 0 \)
So the solutions are x = 2 and x = 3.
Frequently Asked Questions
- How do I know if a system has solutions?
- The system will have solutions if the quadratic equation has real roots and at least one of them satisfies the linear equation.
- What if the quadratic equation doesn't factor nicely?
- The calculator will use the quadratic formula to find solutions when factoring isn't straightforward.
- Can I solve systems with more than two equations?
- This calculator is designed for systems with exactly one quadratic and one linear equation.
- What if the system has no solutions?
- The calculator will indicate when there are no valid solutions that satisfy both equations.
- How accurate are the results?
- The calculator uses precise mathematical methods to solve the equations with floating-point arithmetic.