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The Solving Systems With One Linear And One Quadratic Equation Calculator will calculate:

- The values of the variables x and y in any system of equations where one is linear and the other quadratic.

The first x-solution of the system, x^{1} = |

The second x-solution of the system, x^{2} = |

The first y-solution of the system, y^{1} = |

The second y-solution of the system, y^{2} = |

The first x-solution of the system Formula and Calculations |
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x_{1} = -(a_{1} + b_{1} b_{2} ) - √(a_{1} + b_{1} b_{2} )^{2} - 4a_{2} b_{1} (c_{1} + b_{1} c_{2})/2a_{2} b_{1}x _{1} = -( + × ) - √( + × )^{2} - 4 × × × ( + × )/2 × × x _{1} = - √()^{2} - 4 × × × ()/x _{1} = - √ - 4 × × × /x _{1} = - √/x _{1} = - /x _{1} = /x _{1} = |

The second x-solution of the system Formula and Calculations |

x_{2} = -(a_{1} + b_{1} b_{2} ) + √(a_{1} + b_{1} b_{2} )^{2} - 4a_{2} b_{1} (c_{1} + b_{1} c_{2})/2a_{2} b_{1}x _{2} = -( + × ) + √( + × )^{2} - 4 × × × ( + × )/2 × × x _{2} = + √()^{2} - 4 × × × ()/x _{2} = + √ - 4 × × × /x _{2} = + √/x _{2} = + /x _{2} = /x _{2} = |

The first y-solution of the system Formula and Calculations |

y_{1} = - a_{1} ∙ (a_{1} + b_{1} b_{2} ) + a_{1} ∙ √(a_{1} + b_{1} b_{2} )^{2} - 4a_{2} b_{1} (c_{1} + b_{1} c_{2} ) /2a_{2} b_{1}^{2}c_{1}/b_{1}y _{1} = - ∙ ( + × ) + ∙ √( + × )^{2} - 4 × × × ( + × ) /2 × × ^{2}/y _{1} = - ∙ () + ∙ √()^{2} - 4 × × × () /2 × × y _{1} = - ∙ + ∙ √ - 4 × × × /y _{1} = - ∙ + ∙ √/y _{1} = - ∙ + ∙ /y _{1} = - /y _{1} = - y _{1} = |

The second y-solution of the system Formula and Calculations |

y_{2} = - a_{1} ∙ (a_{1} + b_{1} b_{2} ) - a_{1} ∙ √(a_{1} + b_{1} b_{2} )^{2} - 4a_{2} b_{1} (c_{1} + b_{1} c_{2} ) /2a_{2} b_{1}^{2}c_{1}/b_{1}y _{2} = - ∙ ( + × ) - ∙ √( + × )^{2} - 4 × × × ( + × ) /2 × × ^{2}/y _{2} = - ∙ () - ∙ √()^{2} - 4 × × × () /2 × × y _{2} = - ∙ - ∙ √ - 4 × × × /y _{2} = - ∙ - ∙ √/y _{2} = - ∙ - ∙ /y _{2} = - /y _{2} = - y _{2} = |

Solving Systems With One Linear And One Quadratic Equation Calculator Input Values |

Coefficient preceding x in the linear equation (a_{1}) |

Coefficient preceding y in the linear equation (b_{1}) |

Constant of the linear equation (c_{1}) |

Coefficient preceding x^{2} in the quadratic equation (a_{2}) |

Coefficient preceding x in the quadratic equation (b_{2}) |

Constant of the quadratic equation (c_{2}) |

Please note that the formula for each calculation along with detailed calculations is shown further below this page. As you enter the specific factors of each solving systems with one linear and one quadratic equation calculation, the Solving Systems With One Linear And One Quadratic Equation Calculator will automatically calculate the results and update the formula elements with each element of the solving systems with one linear and one quadratic equation calculation. You can then email or print this solving systems with one linear and one quadratic equation calculation as required for later use.

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3 - x = x^{2} + 2x - 1

orx^{2} + 3x - 4 = 0

where a = 1, b = 3 and c = -4. Hence, using the quadratic formula for solving this new quadratic equation for the variable x yieldsx_{1} = *-b - √***b**^{2} - 4ac*/**2a*

=*-3 - √***3**^{2} - 4 ∙ 1 ∙ (-4)*/**2 ∙ 1*

=*-3 - 5**/**2*

= -4

and=

=

= -4

x_{2} = *-b + √***b**^{2} - 4ac*/**2a*

=*-3 + √***3**^{2} - 4 ∙ 1 ∙ (-4)*/**2 ∙ 1*

=*-3 + 5**/**2*

= 1

The corresponding y-values therefore are=

=

= 1

y_{1} = 3 - x_{1}

= 3 - (-4)

= 7

and = 3 - (-4)

= 7

y_{2} = 3 - x_{2}

= 3 - 1

= 2

In this way, we obtained the two pairs of solutions (-4, 7) and (1, 2) for the given system of equations (as expected). This example corresponds to the cases when the straight line that corresponds to the linear equation graph "pierces" the parabola representing the graph of the quadratic equation, as shown in the figure below. Obviously, this cannot happen always; sometimes the linear graph "touches" the parabola at a single point (it is tangent). This occurs when the discriminant of the new quadratic equation obtained after substituting the linear equation into the quadratic one is zero. Moreover, there are cases when the two graphs do not touch each other at all. This occurs when the discriminant of the abovementioned quadratic equation is negative. Obviously, you don't have to follow such a long procedure as the one described above when solving systems with one linear and one quadratic equation, as our calculator solves the system automatically. You only have to insert the coefficients and constants of the two equations and the result will be displayed automatically. Thus, after making a series of substitutions and transformations in the system derived from the fact that the variable y is expressed in terms of x in the quadratic equation, yields the four formulas below, that give the two possible pairs of solutions (sometimes they show the message "no solution", or the solutions may be equal, depending on the sign of the discriminant). = 3 - 1

= 2

x_{1} = *-(a*_{1} + b_{1} b_{2} ) - √**(a**_{1} + b_{1} b_{2} )^{2} - 4a_{2} b_{1} (c_{1} + b_{1} c_{2})*/**2a*_{2} b_{1}

x_{2} = *-(a*_{1} + b_{1} b_{2} ) + √**(a**_{1} + b_{1} b_{2} )^{2} - 4a_{2} b_{1} (c_{1} + b_{1} c_{2})*/**2a*_{2} b_{1}

y_{1} = *a*_{1} ∙ (a_{1} + b_{1} b_{2} ) + a_{1} ∙ √**(a**_{1} + b_{1} b_{2} )^{2} - 4a_{2} b_{1} (c_{1} + b_{1} c_{2} ) */**2a*_{2} b_{1}^{2} - *c*_{1}*/**b*_{1}

y_{2} = *a*_{1} ∙ (a_{1} + b_{1} b_{2} ) - a_{1} ∙ √**(a**_{1} + b_{1} b_{2} )^{2} - 4a_{2} b_{1} (c_{1} + b_{1} c_{2} ) */**2a*_{2} b_{1}^{2} - *c*_{1}*/**b*_{1}

x

y

y

The following Math tutorials are provided within the Equations section of our Free Math Tutorials. Each Equations tutorial includes detailed Equations formula and example of how to calculate and resolve specific Equations questions and problems. At the end of each Equations tutorial you will find Equations revision questions with a hidden answer that reveal when clicked. This allows you to learn about Equations and test your knowledge of Math by answering the revision questions on Equations.

- 9.1 - Variables, Coefficients and Constants. First Order Equations with One Variable
- 9.2 - Word Problems Involving Equations
- 9.3 - Identities
- 9.4 - Iterative Methods for Solving Equations
- 9.5 - Quadratic Equations
- 9.6 - The Quadratic Formula
- 9.7 - Systems of Linear Equations. Methods for Solving Them.
- 9.8 - Relationship between Equations in Linear Systems. Systems of Equations with One Linear and One Quadratic Equation