Quadratic Equations

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 $x^2 - 2x - 15 = 0$.

Worked Example 2 — Leading Coefficient ≠ 1

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

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.

Worked Example 3

Factor $4x^2 - 25$.

Worked Example 4

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

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

Special Patterns 🎯

Identify the Factoring Method 🔍

For each expression, select the best factoring approach.

Key Takeaways — Part 1

• 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

Quadratic Equations

Part 2 of 7 — The Quadratic Formula & Discriminant

The Quadratic Formula

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

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

Use this when factoring is difficult or impossible.

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

Worked Example 1

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

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

Worked Example 2

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

Worked Example 3

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

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

• 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

Quadratic Equations

Part 3 of 7 — Vertex Form and Completing the Square

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

• Vertex is at $(h, k)$
• $a > 0$: opens up (minimum at vertex)
• $a < 0$: opens down (maximum at vertex)

Converting Standard → Vertex Form (Completing the Square)

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

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

Vertex: $(-3, -7)$

Worked Example 1

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

Worked Example 2 — Leading Coefficient ≠ 1

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

Vertex Form — Basics 🎯

When to Use Each Form

Worked Example 3

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

Worked Example 4

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

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

• Vertex form: $a(x - h)^2 + k$ → vertex at $(h, k)$
• Watch the sign: $(x + 3)$ means $h = -3$
• If $a > 0$: minimum at vertex. If $a < 0$: maximum at vertex
• When $a \neq 1$: factor it out from the $x$-terms first

Quadratic Equations

Part 4 of 7 — Graphing Parabolas

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

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

The Symmetry Trick

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

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

Worked Example 1

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

Worked Example 2

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

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

Worked Example 3

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

Worked Example 4

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

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

• 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

Quadratic Equations

Part 5 of 7 — Quadratic Word Problems

Projectile Motion

The SAT's classic quadratic word problem:

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

• $h_0$ = 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

Worked Example 1

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

Area Problems

"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?

Worked Example 3

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

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

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

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:

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)

Worked Example 1

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

Worked Example 2

For what values of $k$ is the line $y = kx - 1$ tangent to $y = x^2$?

Why the y-intercept matters: a line with a positive y-intercept, such as $y = kx + 2$, can never be tangent to $y = x^2$. Setting them equal gives $x^2 - kx - 2 = 0$ with $\Delta = k^2 + 8$, which is always positive, so the line always crosses the parabola twice.

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 $y = -x^2 + 5$.

Worked Example 4

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

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

Systems with Quadratics 🎯

How Many Intersections? 🔍

Determine the number of intersection points for each system.

Key Takeaways — Part 6

• 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

Quadratic Equations

Part 7 of 7 — SAT Quadratics Review & Hard Problems

Everything You Need to Know

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 $p^2 + q^2$.

Worked Example 2 — Converting Forms

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

Worked Example 3 — Building from Roots

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

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?

Worked Example 5

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

Hard SAT-Style Questions 🎯

Which Strategy? 🔍

Match each SAT question type with the best approach.

Key Takeaways — Part 7 (Full Review)

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.