Solve Quadratic Equations Graphically
This is a tutorial on how to solve quadratic equations graphically and check the answers to the analytical solutions. The quadratic equations explored are of the type
Review
The analytical solutions to the above quadratic equation are given by the quadratic formulaand
x2 = [ - b - √Δ ] / (2a)
- If Δ > 0, the quadratic equation has 2 real solutions.
- If Δ = 0, the quadratic equation has 1 real solution.
- If Δ < 0, the equation has 2 conjugate imaginary solutions.
Question 1
Solve graphically and analytically the equationSolution to Question 1:
Graphical solution
Write the equation in standard form with the right side equal to zero as follows2 x 2 + 3x - 5 = 0
Use any graphing calculator and graph the expression on the left hand side of the equation as a function given by
y = 2 x 2 + 3x - 5
Locate the x intercepts and approximate their x coordinates which are approximations to the solution of the equation.
x 1 = -2.5 and x 2 = 1
![Graphical solution of a quadratic equation with two solutions.](http://www.analyzemath.com/Equations/graph_quadratic_two_real_solutions.gif)
Figure 1. Graphical solution of the quadratic equation 2 x 2 + 3x = 5 with two solutions.
Analytical solution:
Calculate the discriminant Δ and use the quadratic formulas to solve the quadratic equation in standard form.Δ = b 2 - 4ac = 3 2 - 4(2)(-5) = 49
Discriminant Δ is positive, the equation has two real solutions given by.
x 1 = [-3 + √(49)] / (2*2) = 1
x 2 = [-3 - √(49)] / (2*2) = - 2.5
The graphical and analytical solutions are identical. However in general graphical solutions are only approximate.
Question 2
Solve graphically and analytically the equationx 2 + 4x + 4 = 0
Solution to Question 2:
Graphical solution
Use a graphing calculator and graph the functiony = x 2 + 4x + 4
There is one x intercept only at x = - 2 which an approximation to the solution of the given equation.
![Graphical solution of a quadratic equation with one solutions only.](http://www.analyzemath.com/Equations/graph_quadratic_one_solution.gif)
Figure 2. Graphical solution of the quadratic equation x 2 + 4x + 4 = 0 with one solution.
Analytical solution
The discriminant Δ = b2 - 4acΔ = 16 - 4 * 4 = 0
Discriminant Δ is equal to zero, the equation has one solution given by.
x = -b / 2 a = -4 / 2(1) = - 2
The graphical and analytical solutions are equal but in general the graphical solution is approximate.
Question 3
Solve graphically and analytically the equation- x 2 + 4 x - 5 = 0
Solution to Question 3:
Graphical solution
Use a graphing calculator and graph the functiony = - x 2 + 4 x - 5
There are no x intercepts and therefore the solution has no real solution but has complex solutions which are not shown graphically.
![Graphical solution of a quadratic equation with one solutions only.](http://www.analyzemath.com/Equations/graph_quadratic_one_solution.gif)
Figure 3. Graphical solution of the quadratic equation y = - x 2 + 4 x - 5 with no x intercepts.
Analytical solution
-
Given
- x 2 + 4 x - 5 = 0
-
The discriminant Δ = b2 - 4ac
Δ = b2 - 4ac = 42 - 4(-1)(-5) = -4
-
Discriminant Δ is negative, the equation has no real solutions but has two complex conjugate solutions given by.
x1 = [-4 + √(-4)] / (2*(-1)) = 2 - i
x2 = [-4 - √(-4)] / (2*(-1)) = 2 + i
We cannot use the graphical method to find imaginary solutions to an equation.
More Questions
Solve graphically and analytically the following quadratic equations.1: -x 2 - 2 x = 1
2: x 2 + 2 x + 10 = 0
3: x 2 + 2 x = 0
Analytical Solutions to Above Questions
1: graphical: one x intercept at x = -1 , analytical: one double solution -1
2: graphical: no x intercepts , analytical: two imaginary conjugate solutions: -1 - 3i and -1 + 3i
3: graphical: two x intercepts at x = 0 and x = - 2 , analytical: 0 and -2
More References and links
Quadratic Equations Calculator and Solver.