Algebra 1 Workbook — Chapter 15: Solving Quadratic Equations

SOL A.EI.3 (a, b, c) · Companion to Textbook Chapter 15

Each page below is one Canva page. Headings are sized for direct paste: page title as H1, section labels as H2. Figures referenced by filename live in ../figures/. Every numbered item is the same problem as in the textbook, so one answer key serves both. Item numbers run continuously from 1 to 132.

PAGE 1 — Chapter opener

Chapter 15 · Solving Quadratic Equations

Standard A.EI.3 (a, b, c)

In this chapter you will:

Write quadratic equations in standard form and name $a$ , $b$ , and $c$ with their signs
Solve by square roots, factoring, completing the square, and the quadratic formula
Leave irrational answers in simplest radical form
Determine and justify two, one, or no real solutions using the discriminant
Choose a sensible method, verify three ways, and reject physically impossible roots with a reason

Words to know: quadratic equation · standard form · solution · root · zero · $x$ -intercept · square root method · zero product property · completing the square · quadratic formula · discriminant · simplest radical form · no real solutions · verify · interpret · reject

Convention: over the real numbers only. If $b^2 - 4ac < 0$ , write no real solutions — not a complex pair. In context, reject a meaningless root and say why.

PAGE 2 — Standard form anatomy

15.1 Standard Form

FIGURE: fig2-standard-form-anatomy.png (full width)

Fill in the blanks.

Standard form is ____________ = 0, with $a \neq$ ____________.

The sign of $b$ ____________ with the number.

For $3x^2 - 5x + 2 = 0$ : $a =$ ______ $b =$ ______ $c =$ ______

Why is $b$ not $5$ ? _______________________________________________
Write in standard form; name $a$ , $b$ , $c$ .

a) $x^2 = 6x - 8$ → _______________________ $a=$ __ $b=$ __ $c=$ __

b) $2x^2 + 5 = 3x$ → _______________________ $a=$ __ $b=$ __ $c=$ __

c) $4x = x^2 + 3$ → _______________________ $a=$ __ $b=$ __ $c=$ __

PAGE 3 — Solutions on the graph

Checking a Solution on the Graph

FIGURE: fig1-checking-a-solution-on-the-graph.png (full width)

Solutions of $x^2 - x - 6 = 0$ : ____________ and ____________

What feature of the graph names them? _______________________________
Verify both by substitution.

$x$	Substitute into $x^2 - x - 6$	$= 0$ ?
$-2$
$3$

Is $x = 4$ a solution of $x^2 - 5x + 4 = 0$ ? ______ Show: _______________
Explain. Why is a solution the same number as an $x$ -intercept of $y = ax^2 + bx + c$ ?

PAGE 4 — Practice · coefficients and tests

Naming Coefficients and Testing Solutions

Identify $a$ , $b$ , $c$ .

a) $x^2 + 9x - 2 = 0$ → ____________ b) $5x^2 - 3 = 0$ → ____________ c) $-2x^2 + 4x + 1 = 0$ → ____________ d) $x^2 = 0$ → ____________
Standard form and coefficients.

a) $x^2 + 3x = 10$ → ____________ b) $7 = 2x^2 - x$ → ____________ c) $(x - 1)(x + 4) = 0$ → ____________
Test in $x^2 - 5x + 6 = 0$ .

$x$	Substitution	Solution?
$2$
$3$
$-1$
$6$

Graph of $y = x^2 + 2x - 15$ meets axis at $(-5, 0)$ and $(3, 0)$ .

Solutions: ____________ Verify one: ____________

PAGE 5 — Reasoning and application · 15.1

Standard Form in Context

Reasoning. Error in calling $x^2 = 9$ "already standard form":

Correct form: ____________ $a=$ __ $b=$ __ $c=$ __
Error analysis. Student reads $a = 3$ , $b = 5$ , $c = 2$ from $3x^2 - 5x + 2 = 0$ .

Error: ____________ Correct: ____________
Application. Height $h = -16t^2 + 32t + 48$ . Ground: $h = 0$ .

Equation in standard form: _______________________

$a=$ ______ $b=$ ______ $c=$ ______
Reasoning. Why must $a \neq 0$ ? _______________________________________
Sketch $y = x^2 - 4$ ; solutions of $x^2 - 4 = 0$ : ____________
Technology. Window used for $y = x^2 - x - 6$ : ____________

Zeros reported: ____________

PAGE 6 — Exit ticket 15.1

Exit Ticket 15.1

$5x = 2x^2 - 3$ in standard form: ____________ $a=$ __ $b=$ __ $c=$ __
Is $x = -2$ a solution of $x^2 + 3x + 2 = 0$ ? ______ Work: ____________
$y = x^2 - 9$ meets axis at $(-3, 0)$ and $(3, 0)$ . Solutions: ____________
Verify algebraically means: ____________

Verify graphically means: ____________

PAGE 7 — Square roots · two cases

15.2 Solving by Square Roots

FIGURE: fig3-square-roots-two-cases.png (full width)

$x^2 = 9$ : solutions ____________ Left graph shows: ____________

$x^2 = -9$ : conclusion ____________ Right graph shows: ____________
$x^2 = 49$ → ____________
$x^2 - 25 = 0$ → ____________
$3x^2 = 48$ → ____________
$(x - 4)^2 = 9$ → ____________
$x^2 + 7 = 0$ → ____________ Justify: ____________

PAGE 8 — Practice · square roots

Square-Root Practice

Solve.

a) $x^2 = 81$ → ____________ b) $x^2 - 64 = 0$ → ____________ c) $5x^2 = 20$ → ____________ d) $2x^2 - 50 = 0$ → ____________
Solve.

a) $(x + 5)^2 = 16$ → ____________ b) $(x - 3)^2 = 36$ → ____________ c) $(2x)^2 = 100$ → ____________ d) $(x + 1)^2 = 0$ → ____________
Solve or state no real solutions.

a) $x^2 = -4$ → ____________ b) $x^2 + 9 = 0$ → ____________ c) $(x - 2)^2 = -1$ → ____________ d) $4x^2 + 12 = 0$ → ____________
$5(x - 1)^2 - 20 = 0$ → ____________

PAGE 9 — Reasoning · square roots

Errors, Context, and Radicals

Reasoning. Student reports only $x = 4$ for $x^2 = 16$ .

Error: ____________ Complete set: ____________
Error analysis. Why $\sqrt{x^2} = x$ fails for $x^2 = 9$ :
Application. Square patio area $196$ ft $^2$ . Equation: ____________

Solutions: ____________ Keep: ______ Reject ______ because ____________
Verify both solutions of $(x + 2)^2 = 25$ :
Technology. $y = x^2 - 36$ meets axis? ______ Matches $x^2 = 36$ ? ______

$y = x^2 + 16$ meets axis? ______ Matches $x^2 = -16$ ? ______
$(x + 3)^2 = 12$ → ____________ (simplest radical form)

PAGE 10 — Exit ticket 15.2

Exit Ticket 15.2

$4x^2 - 64 = 0$ → ____________
$(x - 5)^2 = 49$ → ____________
$x^2 + 25 = 0$ → ____________
Why no real solutions when $k < 0$ in $x^2 = k$ ?

PAGE 11 — Zero product branch

15.3 Solving by Factoring

FIGURE: fig4-zero-product-branch.png (full width)

Factor: ____________ Branches: ____________ Solutions: ____________
Checks:

$x = -5$ : ____________ $x = 3$ : ____________
Why $(x + 5)(x - 3) = 7$ does not give $x + 5 = 7$ :
$x^2 - 5x + 6 = 0$ → ____________
$x^2 + 7x + 12 = 0$ → ____________
$2x^2 - 5x - 3 = 0$ → ____________

PAGE 12 — Practice · factoring

Factoring Practice

Solve by factoring.

a) $x^2 - 9x + 20 = 0$ → ____________ b) $x^2 + 6x + 8 = 0$ → ____________ c) $x^2 - 3x - 28 = 0$ → ____________ d) $x^2 - 49 = 0$ → ____________
Solve by factoring.

a) $2x^2 + 7x - 4 = 0$ → ____________ b) $3x^2 - 10x - 8 = 0$ → ____________ c) $5x^2 - 5x = 0$ → ____________ d) $4x^2 - 9 = 0$ → ____________
Standard form, then factor.

a) $x^2 = 5x$ → ____________ b) $x^2 + 4x = 21$ → ____________ c) $2x^2 + 3x = 2$ → ____________

PAGE 13 — Factoring reasoning and exit

Factoring · Reasoning and Exit Ticket

Error analysis. $(x - 3)(x + 2) = 6$ solved as $x - 3 = 6$ or $x + 2 = 6$ .

Error: ____________ Standard form: ____________ Solutions: ____________
Reasoning. Why must the product equal zero?

Counterexample: ____________
Application. Consecutive integers, product $72$ .

Equation: ____________ Solutions / pairs: ____________
$x^2 - 8x + 16 = 0$ → ____________ Distinct real solutions: ______
Verify both solutions of $3x^2 + x - 2 = 0$ :
Technology. Zeros of $y = x^2 + 2x - 15$ : ____________
$6x^2 - 7x - 3 = 0$ → ____________
$x^2 - x - 12 = 0$ → ____________
$2x^2 - 7x - 4 = 0$ → ____________
$x(x + 5) = 24$ → standard form ____________ → solutions ____________
Zero product property (one sentence):

PAGE 14 — Completing the square · area

15.4 Completing the Square

FIGURE: fig5-completing-the-square-area.png (full width)

The dashed corner represents: ____________

Why add $9$ to both sides? ____________
Copy the six steps; check both solutions:
$x^2 + 4x - 12 = 0$ → ____________
$x^2 - 8x + 7 = 0$ → ____________
$x^2 + 10x + 21 = 0$ → ____________
For $x^2 + 6x$ : $\left(\dfrac{b}{2}\right)^2 =$ ______ Trinomial: ____________

PAGE 15 — Practice · completing the square

Completing-the-Square Practice

Solve by completing the square.

a) $x^2 + 2x - 15 = 0$ → ____________ b) $x^2 - 6x - 7 = 0$ → ____________ c) $x^2 + 12x + 32 = 0$ → ____________ d) $x^2 - 10x + 24 = 0$ → ____________
Simplest radical form.

a) $x^2 + 4x - 1 = 0$ → ____________ b) $x^2 - 2x - 4 = 0$ → ____________ c) $x^2 + 6x + 2 = 0$ → ____________
$x^2 - 4x + 1 = 0$ → ____________
Reasoning. Why prefer even $b$ when $a = 1$ ?

PAGE 16 — CTS application and exit

Completing the Square · Context and Exit

Error analysis. Student adds $9$ to the left of $x^2 + 6x = 7$ only.

Error: ____________ Correct solutions: ____________
Application. Length is $6$ m more than width $w$ ; area $16$ m $^2$ .

Equation: ____________ Solutions: ____________

Keep: ______ Reject ______ because ____________
Verify both solutions of $x^2 + 8x - 9 = 0$ : ____________
Rewrite $x^2 + 6x - 7$ by completing the square, then solve: ____________
Technology. Zeros of $y = x^2 + 6x - 7$ : ____________
$x^2 + 14x + 40 = 0$ → ____________
$x^2 + 8x - 20 = 0$ → ____________
$x^2 - 2x - 5 = 0$ → ____________
Completes $x^2 - 12x$ : ____________
Missing corner ↔ $\left(\dfrac{b}{2}\right)^2$ : ____________

PAGE 17 — Quadratic formula anatomy

15.5 The Quadratic Formula

FIGURE: fig6-quadratic-formula-anatomy.png (full width)

Three labeled reminders:

$\pm$ : ____________

Discriminant: ____________

$2a$ : ____________
$x^2 - 2x - 3 = 0$ : $D=$ ______ Solutions: ____________
$x^2 - 6x + 9 = 0$ : $D=$ ______ Solution(s): ____________
$x^2 - 2x + 3 = 0$ : $D=$ ______ Conclusion: ____________
$x^2 - 4x - 1 = 0$ → ____________

PAGE 18 — Discriminant panels

Discriminant and $x$ -Axis Crossings

FIGURE: fig7-discriminant-and-x-axis-crossings.png (full width)

Match each panel: crosses / touches / misses.

Left ( $D = 16$ ): ____________ Middle ( $D = 0$ ): ____________ Right ( $D = -8$ ): ____________

Use the irrational-roots figure (next page) after computing here if needed.

PAGE 19 — Irrational roots

Exact and Decimal Roots

FIGURE: fig8-irrational-roots-exact-and-decimal.png (full width)

Exact solutions of $x^2 - 2x - 4 = 0$ : ____________

Decimals (hundredths): ____________ and ____________

PAGE 20 — Formula practice

Formula and Discriminant Practice

Solve with the formula.

a) $x^2 - 5x + 6 = 0$ → ____________ b) $2x^2 + 3x - 2 = 0$ → ____________ c) $x^2 + 6x + 5 = 0$ → ____________ d) $3x^2 - 2x - 1 = 0$ → ____________
$D$ and number of real solutions.

a) $x^2 - 8x + 16 = 0$ → $D=$ __ count: ______ b) $x^2 + 2x + 5 = 0$ → $D=$ __ count: ______ c) $x^2 - 3x - 10 = 0$ → $D=$ __ count: ______ d) $4x^2 - 12x + 9 = 0$ → $D=$ __ count: ______
Simplest radical form.

a) $x^2 - 2x - 4 = 0$ → ____________ b) $x^2 + 4x + 1 = 0$ → ____________ c) $5x^2 + 2x - 1 = 0$ → ____________ d) $2x^2 - 4x - 1 = 0$ → ____________
Reasoning. How $D$ justifies the three panels without naming roots:

PAGE 21 — Formula reasoning and exit

Formula · Context and Exit Ticket

Error analysis. Student uses $b = 5$ in $x^2 - 5x + 2 = 0$ and writes $x = \dfrac{-5 \pm \sqrt{17}}{2}$ .

Error: ____________ Correct: ____________
Application. Area $48$ m $^2$ ; length $2$ m more than width $w$ .

Equation: ____________ Solutions: ____________

Keep: ______ Reject ______ because ____________
Number of real solutions of $3x^2 - 6x + 4 = 0$ : ______ Justify: ____________
$4x^2 - 12x + 9 = 0$ → ____________ Why only one distinct root? ____________
Technology. Approximate zeros of $y = x^2 - 2x - 4$ : ____________
Exact check of $x = 1 + \sqrt{5}$ in $x^2 - 2x - 4 = 0$ :
$2x^2 - 3x - 2 = 0$ → ____________
$x^2 + 2x + 8 = 0$ : $D=$ ______ Conclusion: ____________
$x^2 - 6x + 4 = 0$ → ____________
$D > 0$ : ____________ $D = 0$ : ____________ $D < 0$ : ____________

PAGE 22 — Choosing a method

15.6 Choosing a Method

FIGURE: fig10-choosing-a-solution-method.png (full width)

Method the flow points to, and why.

a)  $x^2 = 20$  → ____________
b)  $x^2 - 5x + 6 = 0$  → ____________
c)  $x^2 + 8x - 9 = 0$  → ____________
d)  $3x^2 - x - 2 = 0$  → ____________

Explain. Why is the formula "the fallback, not the first move"?

_______________________________________________

PAGE 23 — Projectile context

Rejecting a Meaningless Root

FIGURE: fig9-projectile-rejected-root.png (full width)

Algebraic roots: ____________ Keep: ______ Reject ______ because ____________
Verify $t = 3$ : ____________
Application. Same model $h = -16t^2 + 32t + 48$ . When does it hit the ground?

_______________________________________________

Reject: ____________  Algebraic check of kept root: ____________

Error analysis. Student reports both $t = -2$ and $t = 5$ as flight times.

Forgot: ____________

PAGE 24 — Blank grids for verification

Graphical Verification Grids

FIGURE: fig11-blank-grids-for-verification.png (full page)

On grid (a), sketch $y = x^2 - x - 6$ and label both $x$ -intercepts.
On grid (b), mark approximate intercepts for $x = 1 \pm \sqrt{5}$ (use figure 8).
On grid (c), sketch enough of $y = x^2 - 2x + 5$ to show it misses the $x$ -axis.

Use grid (d) for any teacher-assigned check.

PAGE 25 — Method practice and three-way verify

Solve, Choose, Verify

$x^2 + 2x - 15 = 0$ → ____________

Technology confirmation plan: ____________

Choose a method; name it; solve.

a)  $x^2 - 81 = 0$  → method ______  solutions ____________
b)  $x^2 - 9x + 14 = 0$  → method ______  solutions ____________
c)  $x^2 + 6x - 16 = 0$  → method ______  solutions ____________
d)  $2x^2 + 5x - 3 = 0$  → method ______  solutions ____________

Verify three ways for $x^2 + 2x - 15 = 0$ .

Algebra: ____________

Graph: ____________

Technology: ____________

Reasoning. One reason to choose factoring over the formula:

_______________________________________________

PAGE 26 — Context applications

Area, Revenue, and Radicals

Application. Width $x$ , length $x + 4$ , area $45$ .

Equation: ____________  Solutions: ____________

Keep: ______  Reject ______ because ____________

Revenue $R(x) = -x^2 + 40x$ equals $\$300$ .

Equation: ____________  Solutions: ____________

Interpretation: ____________

$(x - 3)^2 = 7$ → method ______ solutions ____________
Garden roots $w = -8$ and $w = 3$ . Keep ______ Reject ______ because ____________

PAGE 27 — Exit ticket 15.6

Exit Ticket 15.6

First method for $x^2 + 10x + 21 = 0$ : ______ Solutions: ____________
Three verification modes: ____________ · ____________ · ____________
$x^2 = -9$ → ____________ Justification: ____________

PAGE 28 — Chapter 15 review · Part A

Chapter 15 Review · Solving

$x^2 - 36 = 0$ → ____________
$x^2 - 8x + 12 = 0$ → ____________
$x^2 + 6x - 7 = 0$ (complete the square) → ____________
$2x^2 - 5x - 3 = 0$ (formula) → ____________
$x^2 - 4x - 2 = 0$ (simplest radical form) → ____________
$x^2 + 4x + 1 = 0$ → method ______ solutions ____________

PAGE 29 — Chapter 15 review · Part B

Chapter 15 Review · Solution Counts

FIGURE: fig7-discriminant-and-x-axis-crossings.png (half width, for reference)

$x^2 - 6x + 9 = 0$ : $D=$ ______ count: ______ Justify: ____________
$x^2 + 2x + 5 = 0$ : $D=$ ______ count: ______ Justify: ____________
$x^2 - 3x - 10 = 0$ : $D=$ ______ count: ______ Solutions: ____________

PAGE 30 — Chapter 15 review · Part C

Chapter 15 Review · Verify and Interpret

FIGURE: fig1-checking-a-solution-on-the-graph.png (half width)

Verify $x^2 - x - 6 = 0$ three ways:

Algebra: ____________

Graph: ____________

Technology: ____________

FIGURE: fig9-projectile-rejected-root.png (half width)

Application. $h = -16t^2 + 32t + 48$ hits ground when?

Method explained: ____________
Application. Area $96$ ft $^2$ ; length $4$ ft more than width.

Equation: ____________ Solutions: ____________

Keep / interpret: ____________

Reject ______ because ____________

Technology check: ____________

Canva production notes

Page size: 8.5 × 11 in, 0.75 in margins
Type: headings 24–28 pt, body 12–14 pt, answer blanks 14 pt with 1.5 line spacing
Item order: a few items are grouped by figure rather than strict numeric order so each page can be answered from the figure on that page (e.g. page 23 pulls 102, 103, 108, 113 with the projectile figure; page 24 uses 104, 111, 110 on blank grids). Numbers still match the textbook; the answer key is in numerical order.
Figure widths: fig1, fig2, fig4, fig5, fig6, fig8, fig9, and fig10 are full width on first appearance. fig3-square-roots-two-cases.png and fig7-discriminant-and-x-axis-crossings.png are multi-panel — set full width and do not place text beside them. fig11-blank-grids-for-verification.png takes a full page. Review pages may repeat fig1, fig7, and fig9 at half width.
Blank work space: reasoning and error-analysis items (6, 11, 12, 14, 20, 31, 32, 40, 43, 50, 51, 60, 61, 70, 71, 80, 81, 90, 91, 106, 113, 114) need at least three ruled lines each. Application items with reject-a-root (33, 72, 92, 108, 109, 115, 131, 132) need two lines per part.
Radical answers: leave stacked radical forms legible; do not force a single underline that smashes $1 + \sqrt{5}$ .
Currency: item 115 uses $300 as plain text outside math delimiters (do not wrap a lone dollar sign in math mode).
Negative signs: use the typographic minus (−), matching the figures, not a hyphen.
No-real-solutions blanks: make the blank long enough for the phrase "no real solutions," not a single number slot — otherwise students invent imaginary answers to fill a short blank.