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

a x 2 + b x + c = 0

Review

The analytical solutions to the above quadratic equation are given by the quadratic formula
x1 = [ - b + √Δ ] / (2a)
and
x2 = [ - b - √Δ ] / (2a)
where Δ = b 2 - 4ac is called the discriminant and gives information about the number and nature of the solutions to quadratic equations. Three possibilities:
  • 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.
The graphical solutions of the above equation are obtained by graphing the left side as a function y = a x 2 + b x + c. The graphical solutions are given by the coordinates of the x intercepts if there are any.

Question 1

Solve graphically and analytically the equation
2 x 2 + 3x = 5

Solution to Question 1:

Graphical solution

Write the equation in standard form with the right side equal to zero as follows
2 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.
x1 = -2.5 and x2 = 1
Graphical solution of a quadratic equation with two solutions.

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 = b2 - 4ac = 32 - 4(2)(-5) = 49
Discriminant Δ is positive, the equation has two real solutions given by.
x1 = [-3 + √(49)] / (2*2) = 1
x2 = [-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 equation

x 2 + 4x + 4 = 0

Solution to Question 2:

Graphical solution

Use a graphing calculator and graph the function
y = 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.

Figure 2. Graphical solution of the quadratic equation x 2 + 4x + 4 = 0 with one solution.

Analytical solution

The discriminant Δ = b2 - 4ac
Δ = 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 function
y = - 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.

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.

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Solve Equations, Systems of Equations and Inequalities.

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