01FOUNDATION
Recognize a quadratic
The highest power of the variable is 2. In standard form, the leading coefficient cannot be zero.
ax² + bx + c
If a = 0, the expression becomes linear.
TOPIC 01 · COMPLETE LESSON
Quadratic
Quadratic
equations.
Recognize the form, choose the shortest path, and connect every solution to its graph and context.
THE CORE MODEL
f(x) = ax² + bx + c
LESSON OUTCOMES
By the end, you can…
- Recognize equivalent quadratic forms
- Choose factoring, graphing, or the formula
- Interpret roots, vertex, and coefficients
- Check whether an answer fits its context
01 · CONCEPT MAP
Eight ideas that work together
Don’t memorize these as isolated formulas. Each representation reveals different information.
02REPRESENT
Read the three forms
Standardax² + bx + c
Vertexa(x − h)² + k
Factoreda(x − r₁)(x − r₂)
03ROOTS
Connect factors to solutions
If a product equals zero, at least one factor must be zero.
(x − 3)(x − 5) = 0
So x = 3 or x = 5. These are also x-intercepts.04ALWAYS WORKS
Use the quadratic formula
Identify a, b, and c—including their signs—before substituting.
x = (−b ± √(b² − 4ac)) / 2a
Use it when a quadratic does not factor cleanly.05PREDICT
Read the discriminant
D = b² − 4ac
D > 0 2 real rootsD = 0 1 real rootD < 0 0 real roots
06TURNING POINT
Locate the vertex
The axis of symmetry passes through the vertex.
x = −b / 2a
Substitute this x-value into the function to find y.07INTERPRET
Translate the graph
Roots show where the output is zero. The vertex represents a maximum or minimum. The sign of a tells the opening direction.
08CONTEXT
Reject impossible answers
A quadratic model may produce two algebraic solutions, but a length, time, or population problem can make one solution invalid.
solve→check units→check domain→answer
02 · DECISION STRATEGY
Choose the shortest valid path
The best method depends on what the question gives and what it asks.
Already factored?
Set each factor equal to zero. Avoid expanding unless the question requires coefficients.
Vertex visible?
Read (h, k) directly from a(x − h)² + k and interpret the maximum or minimum.
Easy integer factors?
Find two numbers with the required product and sum, then apply the zero-product property.
Nothing obvious?
Use a graph or the quadratic formula, then verify the requested value and context.
03 · WORKED EXAMPLE
From equation to meaning
Follow a complete reasoning chain instead of stopping after finding two numbers.
EXAMPLE 1Exam-style reasoning
A projectile’s height is modeled by h(t) = −5t² + 20t + 25, where t is time in seconds. At what positive time does the projectile reach the ground?
1
Ground means height is zero.
−5t² + 20t + 25 = 0
2
Simplify by dividing by −5.
t² − 4t − 5 = 0
3
Factor.
(t − 5)(t + 1) = 0
4
Interpret both solutions.
t = 5 or t = −1
ANSWER5 seconds
Negative time is outside the model’s meaningful domain.
04 · ERROR CONTROL
Four mistakes worth catching
01Losing the sign of bWrite coefficients with signs before substitution.
02Forgetting ±A square-root step often produces two candidates.
03Confusing vertex and rootThe vertex is a turning point; roots are x-intercepts.
04Keeping an impossible valueCheck units, restrictions, and the original context.
05 · ORIGINAL PRACTICE
Check your understanding
Choose an answer, check it, then read the reasoning—not just the letter.
0 of 4 checked
Which expression shows the vertex most directly?
Vertex form a(x − h)² + k displays the vertex (h, k), so the answer is B.
How many real solutions does x² + 4x + 8 = 0 have?
D = 4² − 4(1)(8) = −16. A negative discriminant means no real solutions.
What is the smaller solution to x² − 9x + 20 = 0?
Factor as (x − 4)(x − 5) = 0. The roots are 4 and 5, so the smaller solution is 4.
What is the x-coordinate of the vertex of y = x² − 6x + 11?
x = −b/(2a) = −(−6)/(2·1) = 3.
Represent first. Solve second. Interpret last.
Quadratic questions become easier when you identify what each form reveals and verify the final value against the question’s context.