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# Solving Quadratic Equations Graphically and Algebraically

A quadratic equation is an equation that can be simplified to follow the pattern:

y = a Â· x2 + bÂ· x + c,

where the letter x represents the input, the letter y represents the value of the output and the letters a, b and c are all numbers. Sometimes the numbers a, b and c are referred to as coefficients.

Just as you did for linear and power equations, you solve a quadratic equation when you have been given a y-value and need to find all of the corresponding x-values. For example, if you had been given the quadratic equation:

y = x2 + 8 Â· x +10,

and the y-value,

y = 30,

then solving the quadratic equation would mean finding all of the numerical values of x that work when you plug them into the equation:

x2 + 8 Â· x +10 = 30.

Note that solving this quadratic equation is the same as solving the quadratic equation:

x2 + 8 Â· x +10 - 30 = 30 - 30 (Subtract 30 from each side)

x2 + 8 Â· x - 20 = 0 (Simplify)

Solving the quadratic equation  x2 + 8 Â· x - 20 = 0 will give exactly the same values for x that solving the original quadratic equation,  x2 + 8 Â· x +10 = 30, will give.

The advantage of manipulating the quadratic equation to reduce one side of the equation to zero before attempting to find any values of x is that this manipulation creates a new quadratic equation that can be solved using some fairly standard techniques and formulas.

## Solving a Quadratic Function Graphically Using a Graphing Calculator

When you are trying to solve a quadratic equation of the form:

a Â· x2 + b Â· x + c = 0,

the solutions are the x-coordinates of the points where the graph of the quadratic equation  y = aÂ· x2 + bÂ· x + c cuts the horizontal axis.

From a graphical point of view, the solutions of the manipulated quadratic formula:

x2 + 8 Â· x - 20 = 0,

are the x-values where the graph of  y = x2 + 8 Â· x - 20 cuts the horizontal x-axis. (These points, x = 2 and x = -10, are shown in Figure 1).

Figure 1: Graphical representation of the solutions of a quadratic equation.

There may be zero, one or two places where the graph of the quadratic equation  y = a Â· x2 + bÂ· x + c cuts the horizontal x-axis (see Figure 2, below). This means that the quadratic equation:

a Â· x2 + b Â· x + c = 0,

can have zero, one or two solutions.

You can determine the number of solutions that a quadratic equations has by calculating the discriminant,Δ. The discriminant is equal to:

Δ = b2 - 4 Â· a Â· c .

The sign of the discriminant tells you the number of solutions that exist for the quadratic equation,  a Â· x2 + b Â· x + c = 0.

Figure 2: (a) A quadratic equation with zero solutions. (b) A quadratic equation with exactly one solution. (c) A quadratic equation with exactly two solutions.

 Sign of discriminant,  Δ = b2 - 4 Â· a Â· c Number of solutions of quadratic equation,  a Â· x2 + b Â· x + c = 0 Negative (-) Zero solutions Zero (0) Exactly one solution Positive (+) Exactly two solutions.

Table 1: Determining the number of solutions using the sign of the discriminant.