How to find the roots and vertex of a quadratic on a Casio ClassWiz or fx-CG50
On the fx-991EX: MENU → Equation/Func → 2 (Polynomial) → 2.On the fx-CG50: MENU → EQUATION → F2 (POLYNOMIAL) → F1 (degree 2).
Solve ax² + bx + c = 0 and find the turning point (vertex) of y = ax² + bx + c, to check factorising, the formula or completing the square. Exact keystrokes for the Casio ClassWiz first (the fx-991EX and fx-991CW have different menus, so each has its own tab), then the graphic calculators, a worked example you can check your answers against, the mistakes that cost marks, and the same thing in Desmos.
When you need it
Checking roots you found with the formula, the turning point you found by completing the square, and a quadratic inequality's critical values. The exam wants the algebra too: use the calculator to check.
Practise it: Algebra and functions · IAL P1 algebra.
Keystrokes on your calculator
Pick your calculator: the page remembers it next time.
Quadratic graphs: roots and vertex on the Casio fx-991EX ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991EX ClassWiz: scientific (no graphs).
- MENU, then highlight Equation/Func with the cursor keys and select it (it is the 10th icon).
- 2 (Polynomial), then 2 for ax² + bx + c.
- Type the coefficients: 2 = (−) 8 = 3 =.
- Press = to show x₁, = again for x₂, then keep pressing =: the calculator shows the x and y coordinates of the vertex (a minimum here, a maximum when a is negative).
- Use (−) for −8, not the subtraction key.
Quadratic graphs: roots and vertex on the Casio fx-991CW ClassWiz
Written for the fixed ROM (these models take no OS updates). Casio fx-991CW ClassWiz: scientific (no graphs); the newer ClassWiz with a HOME menu of apps.
- HOME, highlight the Equation app and press OK.
- Choose Polynomial, then ax²+bx+c.
- Type 2, −8 and 3 into the Coefficient Editor (EXE after each).
- EXE shows the solutions one at a time; keep pressing EXE and, after the solutions, the calculator shows the minimum (or maximum) coordinates (x, y) of y = ax² + bx + c: the vertex.
- Press the back key while the last result is shown to return to the Coefficient Editor and change a coefficient.
Quadratic graphs: roots and vertex on the Casio fx-CG50
Written for OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell). Casio fx-CG50: graphic (GDC).
- Roots: MENU, choose EQUATION, F2 (POLYNOMIAL), F1 (degree 2). Type 2, −8, 3 with EXE after each, then F1 (SOLVE).
- Vertex: MENU 5 (GRAPH). Type Y1 = 2X² − 8X + 3 and EXE, then F6 (DRAW).
- SHIFT F5 (G-SOLVE), then F3 (MIN) for the vertex here (F2 (MAX) when a is negative).
- Still in G-SOLVE: F1 (ROOT) gives a root and ► the next one; F4 (Y-ICEPT) gives where the curve crosses the y-axis.
- If MIN or ROOT finds nothing, the point is outside the View Window: SHIFT F3 (V-WINDOW) and widen it.
Quadratic graphs: roots and vertex on the TI-84 Plus CE
Written for OS 5.8.x. TI-84 Plus CE: graphic (GDC); also the TI-84 Plus CE Python / CE-T Python Edition.
- y=: Y1 = 2X² − 8X + 3 (X is the X,T,θ,n key, use (−) for negatives). zoom 6: ZStandard.
- 2nd trace (CALC) → 2: zero. Move left of a root and press enter (Left Bound?), right of it and enter (Right Bound?), then near it and enter (Guess?). Repeat for the other root.
- 2nd trace → 3: minimum (4: maximum when a is negative), with the same Left Bound, Right Bound and Guess. The X= and Y= at the bottom are the vertex.
- minimum and maximum are found numerically, so the last digits can wobble (1.9999998 for 2): round to 3 s.f.
Quadratic graphs: roots and vertex on the TI-Nspire CX II
Written for OS 6.x. TI-Nspire CX II: graphic (GDC), non-CAS; the CX II CAS is not allowed in IB, A Level or IGCSE exams.
- Add a Graphs page and type f1(x) = 2x² − 8x + 3, enter.
- menu → Analyze Graph → Zero. Click to the left of a root (lower bound) and to the right (upper bound). Repeat for the other root.
- menu → Analyze Graph → Minimum (Maximum when a is negative), with lower and upper bounds either side of the vertex.
For teachers: Show on the board Print a keystroke card Teaching GDC skills
Worked example
y = 2x² − 8x + 3. Find (a) the roots of 2x² − 8x + 3 = 0, (b) the coordinates of the vertex.
Answers (checked with Python)
(a) x = 0.418861 and x = 3.58114 (exactly 2 ± √10⁄2, from the formula: the discriminant is 64 − 24 = 40). (b) The vertex is (2, −5): completing the square gives y = 2(x − 2)² − 5. It is a minimum because a = 2 is positive.
Common calculator mistakes
- Typing −8 with the subtraction key instead of the negative key: a Syntax ERROR, or a different equation.
- Giving only the x-coordinate of the vertex when the question asks for its coordinates, or the minimum point when it asks for the minimum value (that is just y).
- Calling the vertex a minimum when a is negative: it is a maximum.
- Writing the decimals when the question says "exact" or "in surd form": use the formula or completing the square by hand.
- A negative discriminant means no real roots: the graph does not cross the x-axis. Do not report a calculator error or complex roots as "the answer" unless the question asks for complex roots.
If you get an error
fx-991EX: Math ERROR, Syntax ERROR, Can't Solve
Math ERROR A result, or a value you typed, is outside the range the calculator can handle, or you asked for something mathematically impossible, such as dividing by zero. Check: Look for a zero denominator, sin⁻¹ or cos⁻¹ of a number bigger than 1 (often a wrong side or angle in the sine or cosine rule), the log of 0 or of a negative number, and a value stored in a variable earlier that you forgot about. Fix: Casio's advice: check the values you typed, use fewer digits and try again. If a variable is the input to a function, make sure its value is one the function accepts.
Syntax ERROR There is a problem with the format of the calculation. In Matrix and Vector calculations it also appears when the sizes do not suit the operation. Check: A missing or extra bracket, or two operation signs in a row. With matrices or vectors: adding a 2 × 2 to a 3 × 3, or a product whose sizes do not match. Fix: Casio's advice: make the corrections. For matrices and vectors, check the size you set for each one.
Can't Solve SOLVE could not find a solution. Check: A typing slip in the equation, or a start value far from the root you want. Fix: Casio's advice: check the equation for mistakes, then give a start value close to the solution you expect.
fx-991CW: Math ERROR, Syntax ERROR
Math ERROR A result, or a value you typed, is outside the range the calculator can handle, or you asked for something mathematically impossible, such as dividing by zero. Check: Look for a zero denominator, sin⁻¹ or cos⁻¹ of a number bigger than 1 (often a wrong side or angle in the sine or cosine rule), the log of 0 or of a negative number, and a value stored in a variable earlier that you forgot about. Fix: Casio's advice: check the values you typed, use fewer digits and try again. If a variable is the input to a function, make sure its value is one the function accepts.
Syntax ERROR There is a problem with the format of the calculation: it is typed in a form the calculator cannot read. Check: A missing or extra bracket, two operation signs in a row, or a function with nothing after it. Fix: Casio's advice: make the corrections and try again.
fx-CG50: Ma ERROR
Ma ERROR Casio's Math ERROR: the result is outside the allowable calculation range, or you are trying to do an illegal mathematical operation, such as dividing by zero. Check: Look for a zero denominator, sin⁻¹ or cos⁻¹ of a number bigger than 1 (often a wrong side or angle in the sine or cosine rule), the log of 0 or of a negative number, and a value stored in a variable earlier that you forgot about. Fix: Casio's advice: check your input values and change them so the result is within the allowable range.
TI-84 Plus CE: ERR:DOMAIN, ERR:INVALID DIM, ERR:SYNTAX
ERR:DOMAIN A value given to a function or instruction is outside the values it accepts. In a regression, TI says it means the data do not suit the model you chose. Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Exponential and power models need positive data. Fix: Change the value that is out of range, or choose a model that suits the data.
ERR:INVALID DIM TI's knowledge base: one or more stat plots are turned on, but the lists they use are empty. Check: A stat plot left on from a statistics question, so the error appears when you graph a function. Fix: Turn off the stat plots you are not using, or put the data back in the lists.
ERR:SYNTAX The command contains a syntax error: a misplaced function, argument, bracket or comma. Check: TI points out that subtract − and negative (−) are different keys, and using the wrong one causes this error. Then check the brackets and commas. Fix: Find the misplaced function, argument, bracket or comma and correct it.
TI-Nspire CX II: Domain error, Syntax error
Domain error An argument must be in a specified domain, and this one is not (TI's example: rand(0)). Check: Is every input allowed? σ must be positive, p between 0 and 1, and sin⁻¹ or cos⁻¹ needs a number from −1 to 1. Fix: Change the value that is out of range.
Syntax error The syntax checker found a problem. Check: A missing bracket or comma. Fix: Correct the expression and try again.
Do it in Desmos
Not in the exam. Desmos is not allowed in A Level exams. Use it to learn, to check a calculator answer, and to see what a function or data set looks like.
- Type y = 2x^2 − 8x + 3.
- Click the curve: the roots and the vertex appear as grey points. Click each one to see its coordinates.
Open desmos.com/calculator (free, in any browser) and type:
y=2x^{2}-8x+3
Questions
Can I use a calculator for quadratic graphs: roots and vertex in the exam?
Yes. Every UK A Level Maths paper (Edexcel 9MA0, AQA 7357, OCR H240) and every Edexcel IAL unit, P1 included, allows a calculator that has no CAS and no wireless link. The fx-CG50 qualifies.
What are the fx-991EX keystrokes for quadratic graphs: roots and vertex?
MENU, then highlight Equation/Func with the cursor keys and select it (it is the 10th icon). Then 2 (Polynomial), then 2 for ax² + bx + c. Then Type the coefficients: 2 = (−) 8 = 3 =. The full steps and the other calculators are in the keystrokes section above.
Does this work on other calculators?
Yes. This page has keystrokes for the Casio fx-991EX ClassWiz, Casio fx-991CW ClassWiz, Casio fx-CG50, TI-84 Plus CE and TI-Nspire CX II. The worked example has the same answers on every model, so use it to check.
What should I write as working?
Write the maths, not the key presses: say what you are finding in normal notation (the equation, the probability statement or the integral), then the calculator's answer to the accuracy the question asks for (often 3 significant figures). Keep full accuracy in later parts.
More calculator skills
What else is free here
- A Level Maths topicsEach topic explained, with a worked example and practice questions.
- Past papers and mark schemesLinks to the boards' own papers, with a timer and topic practice.
- IAL revision notesOne page per topic across P1 to P4, S1 and M1.
- Formula sheets (PDF)Edexcel, AQA, OCR and IAL on one page each, showing what each board's booklet gives you.
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Sources
- Casio fx-991EX / fx-570EX Quick Start Guides : Casio fx-991EX ClassWiz, the fixed ROM (these models take no OS updates).
- Casio fx-991CW / fx-570CW User's Guide : Casio fx-991CW ClassWiz, the fixed ROM (these models take no OS updates).
- Casio fx-CG50 Software User's Guide, OS 3.70 (PDF) : Casio fx-CG50, OS 3.80 (menus unchanged from OS 3.60 onwards as far as we can tell).
- TI-84 Plus CE guidebook (Texas Instruments) : TI-84 Plus CE, OS 5.8.x.
- TI-Nspire CX II guidebooks (Texas Instruments) : TI-Nspire CX II, OS 6.x.
- Desmos Help Center for the Desmos functions.
Menu names are taken from each manufacturer's manual for the models shown. Menus can move slightly between OS updates: if a screen looks different on yours, the manual for your version is the reference, and please tell us so we can update this page. A calculator with no tab here has no checked steps for this skill yet. Worked-example answers were recomputed independently in Python.