The Quadratic Formula

📖 When Factoring Isn't Enough

Factoring is fast — when it works. But many quadratics don't break into nice whole-number factors. The quadratic formula is the tool that solves every quadratic, no matter how ugly the numbers are.

It's printed on the GED formula sheet, so you never have to memorize it. And here's the key point for this lesson: your calculator does every piece of the arithmetic — the −b, the 2a, and the square root. Even when we call it "by hand," you're really doing it on the calculator, one piece at a time. You'll use the TI-30XS MultiView the GED gives you on screen the entire way through.

The dead-giveaway on the GED: when the answer choices contain a ± sign and a square root — like x = 3 ± √5 or x = −5 ± √132 — that's the test telling you to use the quadratic formula. Those answers are only "partially solved" on purpose, because the roots aren't whole numbers.

🧮 The Quadratic Formula

First, the quadratic must be in standard form — everything on one side, equal to zero:

Standard Form

a·x² + b·x + c = 0

a is the number in front of x², b is the number in front of x, and c is the plain number. Keep the signs attached — a minus sign in front of a term belongs to that letter.

The Formula (it's on your formula sheet)

x = −b ± √(b² − 4ac)2a

The ± is why a quadratic has two answers — once you add the square root, once you subtract it. Drop your a, b, and c into the formula and simplify carefully.

Watch the −b. The formula starts with negative b. If b is already negative, then −b becomes positive. Example: if b = −6, then −b = +6.

🛣️ Two Ways to Solve It on the GED

The GED gives you an on-screen TI-30XS MultiView calculator, so you get two calculator-based paths. Pick the one that fits the question.

Method A — The Formula (on the calculator)
Build the formula one piece at a time
Best when the answers show a ± and a √ — only the formula gives those.

Method B — Calculator Table

Enter y = ax² + bx + c, read where y = 0
Best when the answers are whole numbers — fast and no algebra.

How to choose: Glance at the answer choices first. Plain whole numbers → use the Table. A ± with a square root → use the formula.

🔟 Method A — Do It In This Order

Don't try to compute the whole formula at once. Break it into four small calculator moves. The first two are quick — and they often let you eliminate wrong answers before you ever touch the square root.

Step 1 · −b
Find −b on the calculator.
This is the number the answer starts with.

Step 2 · 2a
Find 2a on the calculator.
This is the bottom of the fraction.

Step 3 · Check choices

Compare −b and 2a to the answers.
Wrong denominators / starting numbers drop out — often down to 1–2 left.

Step 4 · The √ part

Last, compute b² − 4ac and its root.
Attach it with the ± to finish.

Why the square root is last: it's the slowest, error-prone piece. By the time you get there, Steps 1–3 have usually narrowed the choices to one or two — so the root is just confirming, not searching.

🔢 Worked Examples

Example 1 — Identify a, b, and c

Identify a, b, and c in: 2x² − 7x + 3 = 0

Step 1: The equation already equals zero, so it's in standard form. Match each term to a, b, c.

a = number on x² = 2

b = number on x, with its sign = −7

c = plain number = 3

a = 2, b = −7, c = 3

Example 2 — The Calculator Order (whole-number answer)

Solve: x² + 5x + 6 = 0 a = 1, b = 5, c = 6

Step 1 — −b: b = 5, so −b = −5. The answer starts with −5.

Step 2 — 2a: 2 × 1 = 2. The whole thing sits over 2.

Step 3 — check the choices: So far x = −5 ± ?2. Any choice that doesn't start with −5 or isn't over 2 is gone.

Step 4 — the √ part (last): b² − 4ac = 5² − 4(1)(6) = 25 − 24 = 1. On the calculator, square-root is the 2nd function above x²:

2nd → √x² then 1 → enter gives 1

So x = −5 ± 12 → −5 + 12 = −2 | −5 − 12 = −3

Answer: x = −2 and x = −3

√1 was a perfect square, so the roots came out whole — Method B (the Table) would also work here.

Example 3 — The ± Radical Answer (the GED giveaway)

Solve: x² − 6x + 4 = 0 a = 1, b = −6, c = 4

Step 1 — −b: b = −6, so −b = +6. Watch the sign: −(−6) = 6.

Step 2 — 2a: 2 × 1 = 2.

Step 3 — check the choices: x = 6 ± ?2. Already you can drop any answer that doesn't begin with 6 (or a reduced 3) over 2.

Step 4 — the √ part (last): b² − 4ac = (−6)² − 4(1)(4) = 36 − 16 = 20.

2nd → √x² then 2 0 → enter gives 4.47…

√20 isn't a whole number, so the answer keeps the radical. Simplify exactly: √20 = √(4·5) = 2√5.

x = 6 ± 2√52 → divide every term by 2 → 3 ± √5.

Answer: x = 3 ± √5

A ± and a √ in the final answer = exactly what the GED choices look like. The decimal 4.47… on your calculator confirms the root isn't whole — so the Table won't land on 0, and the formula is required.

Example 4 — The Calculator Table Method

Solve with the calculator Table: x² + x − 6 = 0

Step 1 — open the table and type the function y = x² + x − 6. The teal x key is the variable key — press it once and it types x:

table → xvar x² + xvar − 6 → enter

Step 2 — set it up: Start = −5, Step = 1, choose Auto (press enter after each). The calculator fills in a table of x and y.

Step 3 — scroll and hunt for y = 0: press the down arrow to move through the rows:

▼ ▼ ▼ …read where y = 0

x	y = x² + x − 6
−4	6
−3	0 ←
−2	−4
−1	−6
0	−6
1	−4
2	0 ←

Answer: x = −3 and x = 2 (the two rows where y = 0)

No algebra needed — the Table is perfect when the answer choices are whole numbers.

⌨️ Using the Table Function — Button by Button

The on-screen GED calculator is the TI-30XS MultiView. Its Table feature turns any quadratic into a list of points so you can spot the roots by eye. The one key students hunt for is the x variable key — here it's shown in teal.

1 · Get to standard form

The equation must equal 0. Whatever sits on the non-zero side is the function you'll type as y.

2 · Open the table

One press — "table" is its own key (not a 2nd function).

table

3 · Type the function — find the x key

The variable key is the tricky one. Press it once and it types x. (Press it again and it cycles to y, z, t — so just one press.) Example for y = x² + 2x − 8:

xvar key x² + 2 xvar key − 8 → enter

Tip: the teal x key sits near the sto→ key. If you ever see y or z appear instead of x, you pressed it too many times — delete and press once.

4 · Accept the defaults — press enter 4 more times

After the function, the calculator asks for Start, Step, and Auto. You don't have to type anything — just keep pressing enter to accept each default and page through to the table.

enter enter enter enter

Counting the enter that locked in your function, that's 5 enters in a row — the fastest way to the table.

5 · Scroll and read where y = 0

Press the down arrow to walk through the rows. Every x with y = 0 is a solution — usually two.

▲ ▼ scroll the list

If you don't see a zero, the roots may be off your range — change Start (try −10) or keep scrolling. If y skips straight from negative to positive and never lands on exactly 0, the answer isn't a whole number — switch to the formula (Method A).

🚫 Common Mistakes

Mistake 1 — Most Common

Mishandling −b when b is negative

If b = −6, then −b = +6, not −6. The formula always flips b's sign first.

Mistake 2

Not setting the equation to zero first

x² + 3x = 10 must become x² + 3x − 10 = 0 before you read off a, b, c.

Mistake 3

Dividing only part of the top by 2a

The whole numerator — both the −b and the root — sits over 2a. Divide everything.

Mistake 4

Forgetting the ± gives two answers

Use + once and − once. Reporting only one root loses half the credit.