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Quadratic Equations

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Quadratic Equations - Complete Interactive Lesson

Part 1: Quadratic Fundamentals

Quadratic Equations

Part 1 of 7 — Standard Form and Factoring

Quadratics are one of the most heavily tested topics on the SAT Math section.

Standard Form: $ax^2 + bx + c = 0$

$a$ determines the direction of the parabola (up if $a > 0$ , down if $a < 0$ )
The vertex is at $x = -\frac{b}{2a}$

Factoring

To factor $x^2 + bx + c$ , find two numbers that multiply to $c$ and add to $b$ .

Example: $x^2 + 7x + 12 = 0$

Numbers that multiply to 12 and add to 7: 3 and 4
$(x + 3)(x + 4) = 0$ → $x = -3$ or $x = -4$

Worked Example 1

Factor and solve $x^2 - 2x - 15 = 0$ .

Step	Work
Find two numbers	Multiply to $-15$ , add to $-2$ : 3 and $-5$
Factor	$(x + 3)(x - 5) = 0$

Worked Example 2 — Leading Coefficient ≠ 1

Factor $2x^2 + 7x + 3 = 0$ .

Step	Work
Multiply $a \cdot c$	$2 \times 3 = 6$
Find numbers	Multiply to 6, add to 7: 1 and 6
Split middle term	$2x^2 + x + 6x + 3$

Zero Product Property

If $ab = 0$ , then $a = 0$ or $b = 0$ . This is why factoring works for solving equations.

Factoring Quadratics 🎯

Special Factoring Patterns

Memorize these — they save significant time on the SAT.

Pattern	Formula	Example
Difference of squares	$a^2 - b^2 = (a+b)(a-b)$

Special Patterns 🎯

Identify the Factoring Method 🔍

For each expression, select the best factoring approach.

Key Takeaways — Part 1

Method	When to Use	Speed
Simple factoring	$a = 1$ , integers	Fastest
AC method	$a \neq 1$	Medium
Difference of squares	pattern

Part 2: Factoring

Quadratic Equations

Part 2 of 7 — The Quadratic Formula & Discriminant

The Quadratic Formula

For $ax^2 + bx + c = 0$ :

$x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

Part 3: Quadratic Formula

Quadratic Equations

Part 3 of 7 — Vertex Form and Completing the Square

Vertex Form: $y = a(x - h)^2 + k$

Vertex is at $(h, k)$

Part 4: Vertex Form

Quadratic Equations

Part 4 of 7 — Graphing Parabolas

Key Features of $y = ax^2 + bx + c$

y-intercept: The point $(0, c)$ — just read the constant

Part 5: Graphing Parabolas

Quadratic Equations

Part 5 of 7 — Quadratic Word Problems

Projectile Motion

The SAT's classic quadratic word problem:

$h(t) = -16t^2 + v_0 t + h_0$

Part 6: Problem-Solving Workshop

Quadratic Equations

Part 6 of 7 — Quadratic Systems and Intersections

Line Meets Parabola

To find where $y = x^2 + 2x - 3$ and $y = x + 1$ intersect:

Part 7: Review & Applications

Quadratic Equations

Part 7 of 7 — SAT Quadratics Review & Hard Problems

Everything You Need to Know

Form	Formula	Best For
Standard	$ax^2 + bx + c$	y-intercept, discriminant
Factored	$a$

2 x^{2} + x + 6 x + 3

Factor $4x^2 - 25$ .

Step	Work
Recognize pattern	$(2x)^2 - 5^2$ → difference of squares
Apply formula	$(2x + 5)(2x - 5)$
Solutions if $= 0$	$x = -5/2$ or $x = 5/2$

Is $x^2 + 8x + 16$ a perfect square trinomial?

Step	Work
Check structure	$a = x$ , is $8x = 2ab$ ? → $b = 4$
Check last term	$b^2 = 16$ ✓
Factor	$(x + 4)^2$

SAT Tip: When you see $x^2 + bx + c$ and $c = (b/2)^2$ , it's a perfect square trinomial.

Standard form: $ax^2 + bx + c$ , vertex at $x = -b/(2a)$
Zero product property: if factors multiply to zero, at least one equals zero
Always double-check by expanding your factored form
Look for GCF before trying other methods

x = \frac{- b \pm b ^{2} - 4 a c}{2 a}

Use this when factoring is difficult or impossible.

The Discriminant: $\Delta = b^2 - 4ac$

Discriminant	# Solutions	Graph
$\Delta > 0$	2 real solutions	Parabola crosses x-axis twice
$\Delta = 0$	1 real solution (double root)	Parabola touches x-axis
$\Delta < 0$	0 real solutions	Parabola doesn't touch x-axis

Solve $3x^2 - 5x + 1 = 0$ using the quadratic formula.

Step	Work
Identify $a, b, c$	$a = 3$ , $b = -5$ , $c = 1$
Discriminant	$(-5)^2 - 4(3)(1) = 25 - 12 = 13$
Apply formula	$x = \frac{5 \pm \sqrt{13}}{6}$
Approximate	$x \approx 1.43$ or $x \approx 0.23$

SAT Favorite Question Type 🎯

"For what values of $k$ does $x^2 + kx + 9 = 0$ have exactly one real solution?"

Set discriminant = 0: $k^2 - 4(1)(9) = 0$ → $k^2 = 36$ → $k = \pm 6$

Quadratic Formula & Discriminant 🎯

When to Use Factoring vs. Quadratic Formula

Situation	Best Method	Why
Simple integers	Factoring	Faster, less error-prone
$a \neq 1$ , messy numbers	Quadratic formula	Guaranteed to work
"How many solutions?"	Discriminant only	Don't need to solve
Non-real answers	Quadratic formula	Factoring won't work

Worked Example 2

Does $5x^2 + 3x + 2 = 0$ have real solutions? If yes, find them.

Step	Work
Discriminant	$3^2 - 4(5)(2) = 9 - 40 = -31$

Worked Example 3

For what value of $k$ does $kx^2 + 12x + 4 = 0$ have exactly one solution?

Step	Work
Set $\Delta = 0$	$12^2 - 4(k)(4) = 0$

SAT Tip: Whenever the SAT says "exactly one solution," "one repeated root," or "tangent to the x-axis," set the discriminant to zero.

Discriminant Deep Dive 🎯

Choose the Best Method 🔍

For each equation, select the most efficient solving approach.

Key Takeaways — Part 2

Tool	What It Tells You
Quadratic formula	The exact values of the solutions
Discriminant ( $\Delta$ )	How many real solutions (0, 1, or 2)
Sum of roots ( $-b/a$ )	Total of solutions without solving
Product of roots ( $c/a$ )	Product of solutions without solving

Quadratic formula: memorize it — it works for ALL quadratics
$\Delta > 0$ : 2 solutions, $\Delta = 0$ : 1 solution, $\Delta < 0$ : 0 real solutions
"Exactly one solution" → set discriminant equal to 0
Factoring is faster when it works — always try it first

a > 0

: opens up (minimum at vertex)

a < 0

: opens down (maximum at vertex)

Converting Standard → Vertex Form (Completing the Square)

Example: $y = x^2 + 6x + 2$

Group: $y = (x^2 + 6x) + 2$
Half of 6 = 3, square it = 9
Add and subtract 9 inside: $y = (x^2 + 6x + 9) - 9 + 2$
Factor: $y = (x + 3)^2 - 7$

Complete the square for $y = x^2 - 8x + 10$ .

Step	Work
Group $x$ terms	$y = (x^2 - 8x) + 10$
Half of $-8$	$-4$
Square it	$16$
Add/subtract	$y = (x^2 - 8x + 16) - 16 + 10$
Factor	$y = (x - 4)^2 - 6$
Vertex	$(4, -6)$ — this is the minimum

Worked Example 2 — Leading Coefficient ≠ 1

Complete the square for $y = 2x^2 + 12x + 7$ .

Step	Work
Factor out $a$ from first two terms	$y = 2(x^2 + 6x) + 7$
Half of 6, squared	$9$
Add/subtract inside	$y = 2(x^2 + 6x + 9 - 9) + 7$
Simplify	$y = 2(x + 3)^2 - 18 + 7$
Final	$y = 2(x + 3)^2 - 11$
Vertex	$(-3, -11)$

Vertex Form — Basics 🎯

When to Use Each Form

Form	Best For	Read Directly
Standard: $ax^2 + bx + c$	y-intercept, discriminant	$c$ = y-intercept
Factored: $a(x - r)(x - s)$	x-intercepts (roots)	$r$ and $s$ = zeros
Vertex: $a(x - h)^2 + k$	Max/min value, vertex	$(h, k)$ = vertex

Worked Example 3

The SAT gives you $f(x) = x^2 - 4x + 7$ . What is the range of $f$ ?

Step	Work
Complete the square	$f(x) = (x-2)^2 - 4 + 7 = (x-2)^2 + 3$

Worked Example 4

Convert $y = 3(x - 2)^2 + 5$ to standard form.

Step	Work
Expand $(x-2)^2$	$x^2 - 4x + 4$

SAT Tip: The SAT may give you vertex form and ask "What is the y-intercept?" Just plug in $x = 0$ : $y = 3(0-2)^2 + 5 = 3(4) + 5 = 17$ .

Vertex Form — Applications 🎯

Match the Form to the Question 🔍

Which form should you convert to in order to answer each question?

Key Takeaways — Part 3

Completing the Square Steps
1. Group $x$ terms: $y = (x^2 + bx) + c$
2. Half of $b$ : $b/2$
3. Square it: $(b/2)^2$
4. Add and subtract: $y = (x^2 + bx + (b/2)^2) - (b/2)^2 + c$
5. Factor the perfect square: $y = (x + b/2)^2 + (c - (b/2)^2)$

Vertex form: $a(x - h)^2 + k$ → vertex at $(h, k)$
Watch the sign: $(x + 3)$ means

x-intercepts (roots/zeros): Set

y = 0

and solve

Vertex:

\left(-\frac{b}{2a},\, f\left(-\frac{b}{2a}\right)\right)

Axis of symmetry:

x = -\frac{b}{2a}

(vertical line through vertex)

Direction: Up if

a > 0

, down if

a < 0

If the roots are at $x = r$ and $x = s$ , then the axis of symmetry is at:

$x = \frac{r + s}{2}$

Sketch the key features of $f(x) = x^2 - 6x + 5$ .

Feature	Calculation	Result
y-intercept	$f(0) = 5$	$(0, 5)$
x-intercepts	$(x-1)(x-5) = 0$	$(1, 0)$ and $(5, 0)$
Axis of symmetry	$x = (1+5)/2$	$x = 3$
Vertex	$f(3) = 9 - 18 + 5$	$(3, -4)$
Direction	$a = 1 > 0$	Opens up

From a graph: a parabola has vertex at $(2, 6)$ and passes through $(0, 2)$ . Find the equation.

Step	Work
Vertex form	$y = a(x - 2)^2 + 6$
Use point $(0, 2)$	$2 = a(0-2)^2 + 6 = 4a + 6$
Solve for $a$	$4a = -4$ → $a = -1$
Equation	$y = -(x - 2)^2 + 6$

Graph Features 🎯

Reading Quadratic Graphs on the SAT

The SAT often shows you a graph and asks questions without giving the equation. Here's what to extract:

Graph Reading Checklist

What They Ask	Where to Look
"For what values is $f(x) > 0$ ?"	Where the graph is ABOVE the x-axis
"For what values is $f(x) < 0$ ?"	Where the graph is BELOW the x-axis
"What is the range?"	From vertex $k$ to $\infty$ (up) or $-\infty$ to $k$ (down)
"How many solutions does $f(x) = 3$ have?"	Draw $y = 3$ and count intersections

Worked Example 3

A parabola has roots at $x = -2$ and $x = 4$ and passes through $(0, -8)$ . Find the vertex.

Step	Work
Factored form	$y = a(x + 2)(x - 4)$
Use $(0, -8)$

Worked Example 4

Where is $f(x) = x^2 - 4$ positive?

Step	Work
Find zeros	$x^2 - 4 = 0$ → $x = \pm 2$
Opens up

SAT Tip: To determine where a parabola is positive/negative, find the roots and use the direction ( $a > 0$ or $a < 0$ ) to decide.

Graph Analysis 🎯

Vertex Location vs. X-Intercepts 🔍

Based on the vertex location and direction, how many x-intercepts does the parabola have?

Key Takeaways — Part 4

Feature	How to Find
Y-intercept	Read $c$ from standard form, or plug $x = 0$
X-intercepts	Factor, quadratic formula, or read from graph
Vertex	$x = -b/(2a)$ , then compute $y$
Axis of symmetry	$x = -b/(2a)$ or midpoint of roots
Direction	$a > 0$ : up, $a < 0$ : down
$f(x) > 0$	Where graph is above x-axis
# of x-intercepts	Sign of discriminant, or vertex position + direction

Vertex below x-axis + opens up = 2 x-intercepts
Vertex above x-axis + opens down = 2 x-intercepts
Vertex on x-axis = 1 x-intercept (tangent)
Vertex on wrong side of x-axis = 0 x-intercepts

$h_0$ = initial height (y-intercept)
$v_0$ = initial velocity
$-16$ accounts for gravity (in feet; use $-4.9$ for meters)

"When does it hit the ground?" → Set $h(t) = 0$ "What is the maximum height?" → Find the vertex

A ball is thrown upward from a 4-foot platform at 48 ft/s. Its height is $h(t) = -16t^2 + 48t + 4$ .

Question	Method	Answer
Initial height?	$h(0) = 4$	$4$ feet
Max height?	Vertex: $t = -48/(2(-16)) = 1.5$	$h(1.5) = -16(2.25) + 72 + 4 = 40$ ft
When hits ground?	$-16t^2 + 48t + 4 = 0$ → quadratic formula	$t \approx 3.08$ seconds

"The length of a rectangle is 3 more than its width. The area is 40. Find the dimensions."

Let width $= w$ . Then $w(w + 3) = 40$ → $w^2 + 3w - 40 = 0$ → $(w + 8)(w - 5) = 0$

Width $= 5$ (reject $-8$ ), length $= 8$ .

Projectile & Area Problems 🎯

Revenue/Profit Optimization

This is a common SAT word problem pattern that combines quadratics with real-world thinking.

Worked Example 2

A theater sells tickets at $20 each and sells 200 tickets. For each $2 price increase, 10 fewer tickets sell. What price maximizes revenue?

Step	Work
Let $x$ = number of $2 increases	Price: $20 + 2x$ , Tickets: $200 - 10x$
Revenue	$R = (20 + 2x)(200 - 10x)$
Expand	$R = 4000 - 200x + 400x - 20x^2 = -20x^2 + 200x + 4000$
Vertex	$x = -200/(2(-20)) = 5$
Optimal price	$20 + 2(5) = \$ 30$
Max revenue	$R = -20(25) + 200(5) + 4000 = \$ 4500$

Worked Example 3

The sum of two numbers is 20. What is the maximum product?

Step	Work
Let one number be $x$	Other number: $20 - x$
Product	$P = x(20 - x) = -x^2 + 20x$

SAT Tip: Optimization problems always lead to finding the vertex. Set up the quadratic, then use $x = -b/(2a)$ .

Optimization & Applications 🎯

Identify the Approach 🔍

For each word problem, select the key equation setup.

Key Takeaways — Part 5

Problem Type	Setup	Key Step
Projectile	$h = -16t^2 + v_0t + h_0$	Vertex for max, $h = 0$ for landing
Area	Length × Width = Area	Set up quadratic, reject negatives
Revenue	Price × Quantity	Both depend on same variable; vertex = max
Max product	$P = x(S - x)$ where $S$ = sum	Vertex gives equal values

Always re-read the question: "When?" ≠ "What height?"
Reject negative solutions for time, length, width
"Maximize" or "optimize" = find the vertex

Set equal: $x^2 + 2x - 3 = x + 1$ → $x^2 + x - 4 = 0$

Solve for $x$ , then plug back in for $y$ .

Number of Intersections

The discriminant of the resulting equation tells you:

$\Delta > 0$ : 2 intersection points
$\Delta = 0$ : 1 point (line is tangent to parabola)
$\Delta < 0$ : 0 points (no intersection)

Find where $y = x^2 - 3x + 2$ and $y = x - 1$ intersect.

Step	Work
Set equal	$x^2 - 3x + 2 = x - 1$
Rearrange	$x^2 - 4x + 3 = 0$
Factor	$(x - 1)(x - 3) = 0$
Solve	$x = 1$ or $x = 3$
Find $y$ -values	$y(1) = 0$ , $y(3) = 2$
Intersections	$(1, 0)$ and $(3, 2)$

For what value of $k$ is $y = kx + 2$ tangent to $y = x^2$ ?

Step	Work
Set equal	$x^2 = kx + 2$ → $x^2 - kx - 2 = 0$
Tangent → $\Delta = 0$	$k^2 - 4(1)(-2) = 0$ →
Wait — $k^2 = -8$ ?	No real solution! The line $y = kx + 2$ (y-int $= 2$ ) can't be tangent to

Let's try $y = kx - 2$ instead: $x^2 - kx + 2 = 0$ , $\Delta = k^2 - 8 = 0$ → $k = \pm 2\sqrt{2}$ ✓

Line-Parabola Intersections 🎯

Two Parabolas Intersecting

Set the equations equal: $f(x) = g(x)$ , rearrange to standard form, then solve.

Worked Example 3

Find the intersection(s) of $y = x^2 + 1$ and $y = -x^2 + 5$ .

Step	Work
Set equal	$x^2 + 1 = -x^2 + 5$
Rearrange	→

Worked Example 4

$y = c$ intersects $y = x^2 - 4$ at exactly one point. What is $c$ ?

Step	Work
Set equal	$x^2 - 4 = c$ → $x^2 = c + 4$

SAT Tip: A horizontal line $y = c$ intersects $y = ax^2 + bx + c'$ at exactly one point when equals the -coordinate of the vertex.

Systems with Quadratics 🎯

How Many Intersections? 🔍

Determine the number of intersection points for each system.

Key Takeaways — Part 6

System Type	Method	# of Solutions
Line + Parabola	Set equal, get quadratic, check $\Delta$	0, 1, or 2
Two parabolas	Set equal, simplify	0, 1, or 2
Horizontal line + parabola	$c = ax^2 + bx + c'$ → solve	Depends on $c$ vs vertex

To find intersections: set equal → rearrange → solve
Tangent = 1 intersection = $\Delta = 0$
A horizontal line through the vertex gives exactly 1 intersection
Parallel parabolas (same $a$ , different $c$ ) never intersect

Sum and Product of Roots (Vieta's Formulas)

For $ax^2 + bx + c = 0$ with roots $r$ and $s$ :

Sum: $r + s = -b/a$
Product: $r \cdot s = c/a$

This saves time when the SAT asks for $r + s$ or $rs$ without needing individual roots.

Worked Example 1 — Vieta's Shortcut

The equation $2x^2 - 10x + 7 = 0$ has roots $p$ and $q$ . Find $p^2 + q^2$ .

Step	Work
Sum of roots	$p + q = -(-10)/2 = 5$
Product of roots	$pq = 7/2$
Identity	$p^2 + q^2 = (p+q)^2 - 2pq$
Substitute	$= 25 - 7 = 18$

Worked Example 2 — Converting Forms

Write $2x^2 + 12x + 7$ in vertex form.

Step	Work
Factor out $a$ from $x$ -terms	$2(x^2 + 6x) + 7$
Complete the square inside	$2(x^2 + 6x + 9 - 9) + 7$
Simplify	$2(x + 3)^2 - 18 + 7$
Final answer	$2(x + 3)^2 - 11$

Worked Example 3 — Building from Roots

A quadratic has roots $3$ and $-5$ and passes through $(1, -24)$ . Find the equation.

Step	Work
Start with factored form	$y = a(x - 3)(x + 5)$
Plug in $(1, -24)$	$-24 = a(1-3)(1+5) = a(-2)(6)$
Solve for $a$	$-24 = -12a$ → $a = 2$
Final answer	$y = 2(x - 3)(x + 5) = 2x^2 + 4x - 30$

Vieta's Formulas & Form Conversions 🎯

Hard SAT Patterns

Pattern 1: "The equation has no real solutions"

This means $\Delta < 0$ . Set up $b^2 - 4ac < 0$ and solve for the unknown.

Pattern 2: Nested expressions

"If $x^2 + 3x = 7$ , what is $x^2 + 3x + 5$ ?"

Don't solve for $x$ ! Just substitute: $7 + 5 = 12$ .

Worked Example 4

For what values of $k$ does $kx^2 + 6x + k = 0$ have two distinct real roots?

Step	Work
Need $\Delta > 0$	$36 - 4(k)(k) > 0$
Simplify

Worked Example 5

If $2x^2 - 5x + 1 = 0$ , find $\frac{1}{r} + \frac{1}{s}$ where are the roots.

Step	Work
Rewrite	$\frac{1}{r} + \frac{1}{s} = \frac{r + s}{rs}$

Hard SAT-Style Questions 🎯

Which Strategy? 🔍

Match each SAT question type with the best approach.

Key Takeaways — Part 7 (Full Review)

Concept	Key Formula / Idea
Standard form	$ax^2 + bx + c$ — gives $y$ -int ( $c$ ) and discriminant
Vertex form	$a(x-h)^2 + k$ — gives vertex and max/min
Factored form	$a(x-r)(x-s)$ — gives roots directly
Vieta's: sum	$r + s = -b/a$
Vieta's: product	$r \cdot s = c/a$
Discriminant	$\Delta = b^2 - 4ac$ — 2, 1, or 0 real roots
Vertex $x$ -coordinate	$x = -b/(2a)$
Completing the square	Factor out $a$ , add & subtract $(b/2a)^2$

Final SAT Tip: Before solving, ask: "What is the question actually asking for?" Often you can use Vieta's or substitution without finding individual roots.

a > 0

: minimum at vertex. If

a < 0

: maximum at vertex

When

a \neq 1

: factor it out from the

x

-terms first

Quadratic Equations

Quadratic Equations - Complete Interactive Lesson

Part 1: Quadratic Fundamentals

Quadratic Equations

Standard Form: ax2+bx+c=0ax^2 + bx + c = 0ax2+bx+c=0

Factoring

Worked Example 1

Worked Example 2 — Leading Coefficient ≠ 1

Zero Product Property

Special Factoring Patterns

Key Takeaways — Part 1

Part 2: Factoring

Quadratic Equations

The Quadratic Formula

Part 3: Quadratic Formula

Quadratic Equations

Vertex Form: y=a(x−h)2+ky = a(x - h)^2 + ky=a(x−h)2+k

Part 4: Vertex Form

Quadratic Equations

Key Features of y=ax2+bx+cy = ax^2 + bx + cy=ax2+bx+c

Part 5: Graphing Parabolas

Quadratic Equations

Projectile Motion

Part 6: Problem-Solving Workshop

Quadratic Equations

Line Meets Parabola

Part 7: Review & Applications

Quadratic Equations

Everything You Need to Know

Worked Example 3

Worked Example 4

The Discriminant: Δ=b2−4ac\Delta = b^2 - 4acΔ=b2−4ac

Worked Example 1

SAT Favorite Question Type 🎯

When to Use Factoring vs. Quadratic Formula

Worked Example 2

Worked Example 3

Key Takeaways — Part 2

Converting Standard → Vertex Form (Completing the Square)

Worked Example 1

Worked Example 2 — Leading Coefficient ≠ 1

When to Use Each Form

Worked Example 3

Worked Example 4

Key Takeaways — Part 3

The Symmetry Trick

Worked Example 1

Worked Example 2

Reading Quadratic Graphs on the SAT

Graph Reading Checklist

Worked Example 3

Worked Example 4

Key Takeaways — Part 4

Worked Example 1

Area Problems

Revenue/Profit Optimization

Worked Example 2

Worked Example 3

Key Takeaways — Part 5

Number of Intersections

Worked Example 1

Worked Example 2

Two Parabolas Intersecting

Worked Example 3

Worked Example 4

Key Takeaways — Part 6

Sum and Product of Roots (Vieta's Formulas)

Worked Example 1 — Vieta's Shortcut

Worked Example 2 — Converting Forms

Worked Example 3 — Building from Roots

Hard SAT Patterns

Pattern 1: "The equation has no real solutions"

Pattern 2: Nested expressions

Worked Example 4

Worked Example 5

Key Takeaways — Part 7 (Full Review)

Standard Form: $ax^2 + bx + c = 0$

Vertex Form: $y = a(x - h)^2 + k$

Key Features of $y = ax^2 + bx + c$

The Discriminant: $\Delta = b^2 - 4ac$