How to solve a quadratic equation and find its extremum

Enter three coefficients to find both the roots and the vertex of a quadratic function.

5 steps

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Use Polynomial equation mode after writing the equation in the form:

a x^{2} + b x + c = 0

This example solves:

x^{2} - 5 x + 6 = 0

The three coefficients are $a = 1$ , $b = - 5$ , and $c = 6$ . The calculator asks for them from the highest power down to the constant, so you do not enter the variable $x$ or the equals sign.

Enter the coefficients, find the roots, and read the extremum

Below each step is the screen you should see after pressing.

Next to each step is the screen you should see after pressing.

Step 1

Open the mode picker. On the 991CW-style keypad, choose HOME → Equation → Polynomial → degree 2; on other keypads, choose Polynomial equation → degree 2. The coefficient row appears with $a$ selected.

Press:
Step 2

Enter $1$ for coefficient $a$ , then press the equals key to move to coefficient $b$ .

Press:
Step 3

Enter $- 5$ for coefficient $b$ (for example, with the negative-sign key), then press the equals key to move to coefficient $c$ .

Press:
Step 4

Enter $6$ for coefficient $c$ and press the equals key. Because this is the last coefficient, the calculator solves the equation and shows both roots.

Press:
Step 5

Continue to the Minimum section below the roots. The calculator shows $x = \frac{5}{2}$ and $y = - \frac{1}{4}$ , the coordinates of the parabola's lowest point.

The two results you get

The result for this example is:

x_{1} = 2, x_{2} = 3

At the same time, for the function $y = x^{2} - 5 x + 6$ , the calculator gives the minimum point:

I (\frac{5}{2}, - \frac{1}{4})

Because $a = 1 > 0$ , the parabola opens upward, so this is a minimum. When $a < 0$ , the parabola opens downward and the calculator labels the point Maximum. This is the advantage of quadratic-equation mode: the same three coefficients give you both the equation's roots and the function's extremum.

If the result is wrong

Put the equation into the form $a x^{2} + b x + c = 0$ before reading off the coefficients.

The fx-991EX screen showing the roots x = 2 and x = 3 of x² − 5x + 6 = 0.

Solve a quadratic equation on your fx-991EX calculator

Enter x² − 5x + 6 = 0 and get both roots plus the vertex in one pass — it takes only a few seconds.

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Enter the coefficients, find the roots, and read the extremum

Step 1

Step 2

Step 3

Step 4

Step 5

The two results you get

If the result is wrong

Solve a quadratic equation on your fx-991EX calculator