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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 $x = -4$

Factoring with leading coefficient ≠ 1

For $2x^2 + 7x + 3$ :

Multiply $a \cdot c = 6$ . Find numbers that multiply to 6 and add to 7: 1 and 6
Split: $2x^2 + x + 6x + 3 = x(2x + 1) + 3(2x + 1) = (x + 3)(2x + 1)$

Zero Product Property

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

Factoring Quadratics 🎯

Key Takeaways — Part 1

Standard form: $ax^2 + bx + c$ , vertex at $x = -b/(2a)$
Factoring: find two numbers that multiply to and add to (when )

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	$a$

Zero product property: if the factors multiply to zero, at least one factor equals zero

Always double-check by expanding your factored form

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

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 🎯

Key Takeaways — Part 2

Quadratic formula: memorize it — it works for ALL quadratics
Discriminant ( $b^2 - 4ac$ ) tells you HOW MANY solutions without solving
$\Delta > 0$ : 2 solutions, $\Delta = 0$ : 1 solution, $\Delta < 0$ : 0 real solutions
"Exactly one solution" → set discriminant equal to 0

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: $y = (x + 3)^2 - 7$

When to Use Each Form

Form	Best For
Standard: $ax^2 + bx + c$	y-intercept (just read $c$ ), discriminant
Factored: $a(x - r)(x - s)$	x-intercepts (roots are $r$ and $s$ )
Vertex: $a(x - h)^2 + k$	Maximum/minimum value, vertex

Vertex Form 🎯

Key Takeaways — Part 3

Vertex form: $a(x - h)^2 + k$ → vertex at $(h, k)$
Watch the sign: $(x + 3)$ means $h = -3$
Completing the square: half the $b$ -coefficient, square it, add/subtract
Use vertex form when the SAT asks for minimum, maximum, or "what is the least possible value"

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:

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

This is the midpoint of the roots!

SAT Graph Reading Skills

When the SAT shows you a parabola:

The vertex tells you the min/max
The x-intercepts are the solutions to $f(x) = 0$
The y-intercept is $f(0)$
$f(x) > 0$ is where the graph is above the x-axis
$f(x) < 0$ is where the graph is below the x-axis

Parabola Graphs 🎯

Key Takeaways — Part 4

Read the y-intercept directly from the constant $c$
X-intercepts come from factoring or the quadratic formula
Axis of symmetry = midpoint of roots = $-b/(2a)$
Opens up ( $a > 0$ ) → vertex is minimum; opens down ( $a < 0$ ) → vertex is maximum
" $f(x) > 0$ " means "where is the graph above the x-axis?"

$h_0$ = 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

"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 $= 8$ .

Revenue/Profit Problems

"A store sells 100 items at $20 each. For every $1 increase in price, 5 fewer items sell."

Revenue $= (20 + x)(100 - 5x) = -5x^2 + 0x + 2000$

Maximum revenue at vertex: $x = 0/(2 \cdot (-5)) = 0$ ... meaning $20 is already optimal!

Quadratic Word Problems 🎯

Key Takeaways — Part 5

Projectile: $h(t) = -16t^2 + v_0 t + h_0$ — max height at vertex, hits ground at $h = 0$
Area problems: set up the equation, expand, solve (reject negative answers for dimensions)
Revenue: $R = (\text{price})(\text{quantity})$ — max revenue at vertex of the quadratic
Always re-read the question: "When does it hit the ground?" ≠ "What is the max height?"

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)

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

Rearrange to get a quadratic in standard form, then solve.

The SAT often asks: "At which point(s) does the line $y = c$ intersect $y = x^2 - 4$ ?"

This is just solving $x^2 - 4 = c$ → $x^2 = c + 4$ → $x = \pm\sqrt{c + 4}$

Two solutions when $c > -4$ , one when $c = -4$ , none when $c < -4$ .

Quadratic Systems 🎯

Key Takeaways — Part 6

To find intersections: set the equations equal, solve the resulting quadratic
The discriminant of the combined equation → number of intersection points
"Tangent" = exactly one intersection = discriminant equals zero
$y = c$ intersects $y = x^2 + k$ : solve $x^2 + k = c$ to find $x$ values

Sum and Product of Roots

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 asking for individual roots.

Hard SAT Pattern: Equivalent Forms

"Which is equivalent to $2x^2 + 12x + 7$ ?"

Complete the square: $2(x^2 + 6x) + 7 = 2(x^2 + 6x + 9 - 9) + 7 = 2(x + 3)^2 - 11$

Hard SAT Pattern: Creating Equations

"A quadratic has roots 3 and $-5$ ." → $y = (x - 3)(x + 5) = x^2 + 2x - 15$

Advanced Review 🎯

Key Takeaways — Part 7

Sum of roots $= -b/a$ , Product of roots $= c/a$ (Vieta's formulas)
Know all three forms and when each is most useful
Completing the square: factor out $a$ first, then complete inside the parentheses
To create a quadratic from roots: $y = a(x - r)(x - s)$ , find $a$ from another point

Quadratic Equations

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

SAT Favorite Question Type 🎯

Key Takeaways — Part 2

Converting Standard → Vertex Form (Completing the Square)

When to Use Each Form

Key Takeaways — Part 3

The Symmetry Trick

SAT Graph Reading Skills

Key Takeaways — Part 4

Area Problems

Revenue/Profit Problems

Key Takeaways — Part 5

Number of Intersections

Two Parabolas

SAT Tip 💡

Key Takeaways — Part 6

Sum and Product of Roots

Hard SAT Pattern: Equivalent Forms

Hard SAT Pattern: Creating Equations

Key Takeaways — Part 7

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

Factoring with leading coefficient ≠ 1

Zero Product Property

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

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

SAT Favorite Question Type 🎯

Key Takeaways — Part 2

Converting Standard → Vertex Form (Completing the Square)

When to Use Each Form

Key Takeaways — Part 3

The Symmetry Trick

SAT Graph Reading Skills

Key Takeaways — Part 4

Area Problems

Revenue/Profit Problems

Key Takeaways — Part 5

Number of Intersections

Two Parabolas

SAT Tip 💡

Key Takeaways — Part 6

Sum and Product of Roots

Hard SAT Pattern: Equivalent Forms

Hard SAT Pattern: Creating Equations

Key Takeaways — Part 7

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