← Solving Systems of Equations Including Quadratics

Method

Solving Nonlinear Systems Graphically

If the equations can be graphed, the solutions of a nonlinear system are the points of intersection of the graphs of every equation that makes the system. Therefore, by finding the points of intersection of the graphs, it is possible to solve a nonlinear system. As an example, consider a linear-quadratic system.

{2 x - 2 = y 2 x^{2} - 8 x + 6 = y

To find the solutions, first each equation is graphed.

Graphing the First Equation

The first equation of the nonlinear system is a linear equation written in slope-intercept form. By using the slope of $2$ and the $y -$ intercept of $- 2,$ the equation is graphed in a coordinate plane.

Then, the second equation needs to be graphed.

Graphing the Second Equation

The second equation is a quadratic equation written in standard form.

a x^{2} + b x + c = y ⇓ 2 x^{2} + (- 8) x + 6 = y

To graph a quadratic equation in standard form, first the axis of symmetry needs to be identified and graphed. To do so, the values are substituted into the formula.

x = - \frac{b}{2 a}

SubstituteII

$a = 2$ , $b = - 8$

x = - \frac{- 8}{2 ( 2 )}

▼

Solve for

x

Multiply

x = - \frac{- 8}{4}

CalcQuot

Calculate quotient

x = - (- 2)

NegNeg

$- (- a) = a$

x = 2

This indicates that the axis of symmetry is a vertical line on

x = 2 .

Then, to find the vertex of the parabola, the equation is evaluated to find the value of

y

when

x = 2 .

2 x^{2} - 8 x + 6 = y

Substitute

$x = 2$

2 (2)^{2} - 8 (2) + 6 = y

▼

Solve for

y

CalcPow

Calculate power

2 (4) - 8 (2) + 6 = y

Multiply

8 - 16 + 6 = y

AddTerms

Add terms

- 2 = y

RearrangeEqn

Rearrange equation

y = - 2

This means that the vertex lies on point

(2, - 2) .

Then, the $y -$ intercept is the value of $c$ of the equation, which in this case is $6 .$ Since the axis of symmetry divides the graph into two mirror images, the parabola also goes through point in $(4, 6) .$

Using these points, it is possible to draw the parabola.

Finding the Points of Intersection

Finally, finding the points in which the graph intersect, the solutions of the nonlinear system are found.

Linear Equation and Quadratic Equation Intersection

This means that the points $(1, 0)$ and $(4, 6)$ are the solutions of the nonlinear system.

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