006-MWALA_LEARN-QUADRATIC-SOLVING-FOREVER-2

Objectives: QUADRATIC EQUATIONS

Mastering Quadratic Equations — Complete Notes (Everything You Need)

Mastering Quadratic Equations — Complete Notes (Everything You Need)

From zero to exam-ready: definitions, derivations, every formula (with symbols explained), techniques, real-life examples, and long worked solutions.

1) What is a Quadratic?

A quadratic equation is any equation that can be written in the form a x^2 + b x + c = 0 where a, b, c are real (or complex) constants and a ≠ 0.

Its graph (when written as y = a x^2 + b x + c) is a parabola.

Real-life feel:

  • Projectile height over time (ignoring air): h(t) = -4.9 t^2 + v_0 t + h_0.
  • Profit vs. price often forms a concave parabola.
  • Area problems under fixed perimeter produce quadratics.
Exam pointer: If the highest power of x is 2, think “quadratic.” Check if you can rearrange to ax^2+bx+c=0.

2) Notation & Symbol Glossary

SymbolMeaning
a, b, cReal constants (coefficients). a ≠ 0.
xUnknown variable (what we solve for).
Δ or DDiscriminant: b^2 − 4ac.
x_1, x_2Solutions/roots/zeros of the equation.
V(h, k)Vertex of the parabola (h is x-coordinate, k is y-coordinate).
AxisAxis of symmetry: vertical line x = -b/(2a).
Interceptsy-intercept at (0, c). x-intercepts are the real roots (if any).

3) Forms of a Quadratic & Conversions

A) Standard (General) Form

y = a x^2 + b x + c

Most universal; coefficients are obvious. Good for using the quadratic formula, discriminant, and sum/product of roots.

B) Vertex Form

y = a (x - h)^2 + k

Vertex is V(h, k). Great for graphing and optimization (max/min occurs at x = h).

Convert from standard by completing the square (see §5C).

C) Factored (Intercept) Form

y = a (x - r_1)(x - r_2)

Shows roots directly (x = r_1, r_2 when real). Great for quick graphing and solving.

Conversion map:

Standard → Vertex Complete the square.

Standard → Factored Factor (if possible) or use roots from formula then write a(x-r_1)(x-r_2).

Vertex → Standard Expand.

Factored → Standard Expand; c = a(-r_1)(-r_2) = a r_1 r_2.

4) Graphing Parabolas (What Every Point Means)

  • Opening direction: a > 0 opens up (min). a < 0 opens down (max).
  • Axis of symmetry: x = -b/(2a).
  • Vertex: V\big(-b/(2a),\; f(-b/(2a))\big) or read (h,k) from vertex form.
  • y-intercept: (0, c).
  • x-intercepts (zeros): where y=0, solve the quadratic.
  • Stretch/Shrink: |a| bigger → narrower; |a| smaller → wider.
Real-life: Vertex gives max height of a ball; axis gives the time exactly in the middle of going up and down.

5) Core Methods to Solve Quadratics

A) Factoring

Rewrite ax^2 + bx + c = 0 as (px + q)(rx + s) = 0. Then each factor equals zero.

  • Look for common factor first.
  • For a=1, find numbers that multiply to c and add to b.
  • For a≠1, use “AC method”: find two integers whose product is a·c and sum to b, then group.
Example: Solve x^2 - 5x + 6 = 0.

Numbers multiplying to 6 and adding to −5 → −2 and −3.

x^2 - 5x + 6 = (x-2)(x-3)=0 ⇒ x=2,3.

Watch: Not all quadratics factor nicely over integers. Have other methods ready.

B) Taking Square Roots (When No x term)

Use when in form a(x - h)^2 = k.

Rule: If u^2 = m, then u = ±√m.

Example: 2(x-3)^2 = 18 ⇒ (x-3)^2 = 9 ⇒ x-3=±3 ⇒ x=6 or 0.

C) Completing the Square

Goal: turn ax^2 + bx + c into a(x-h)^2 + k.

  1. Make the x^2 coefficient 1 (divide both sides by a if needed).
  2. Move constant to the other side.
  3. Add (\tfrac{b}{2})^2 (or (\tfrac{b}{2a})^2 before dividing) to both sides to make a perfect square.
  4. Factor left as (x + \tfrac{b}{2})^2, then solve by square roots.
Example: Solve x^2 + 6x - 7 = 0.
Move -7: x^2 + 6x = 7.
Add (6/2)^2 = 9 to both sides: x^2 + 6x + 9 = 16.
Left is (x+3)^2: so (x+3)^2 = 16 ⇒ x+3 = ±4 ⇒ x = 1 or -7.

D) Quadratic Formula (Universal Method)

x = \dfrac{-b ± \sqrt{b^2 - 4ac}}{2a} (derive in §6). Works for any quadratic (a≠0).

Example: Solve 2x^2 + 3x - 2 = 0.
a=2, b=3, c=-2 → Δ = 3^2 - 4·2·(-2) = 9+16=25.
x = (-3 ± √25)/(4) = (-3 ± 5)/4 ⇒ x=1/2 or x=-2.
Technique: Always compute Δ first to see difficulty and root types.

6) Quadratic Formula — Derivation, Symbols, and Why It Works

Start: ax^2 + bx + c = 0, with a ≠ 0.

  1. Divide by a: x^2 + (b/a)x + c/a = 0.
  2. Move constant: x^2 + (b/a)x = -c/a.
  3. Complete the square: add (\tfrac{b}{2a})^2 both sides.
    Left: x^2 + (b/a)x + (b/2a)^2 = (x + b/(2a))^2.
    Right: -c/a + (b^2)/(4a^2) = (b^2 - 4ac)/(4a^2).
  4. Take square roots: x + b/(2a) = ± √(b^2 - 4ac)/(2a).
  5. Isolate x: x = [-b ± √(b^2 - 4ac)]/(2a).
Symbols explained:
  • a, b, c: the coefficients from the original quadratic.
  • Δ = b^2 - 4ac: the part under the square root, called the discriminant.
  • ±: there are two solutions, one with plus and one with minus (unless the root is repeated).
  • 2a: comes from taking square roots after making a perfect square; it normalizes the result.
Why it works (intuition): Any quadratic can be turned into a perfect square by adding just the right amount. Solving the square gives two symmetric solutions around the axis x = -b/(2a). The formula is exactly that process written once and for all.

7) The Discriminant (Δ = b^2 - 4ac)

CaseMeaning for rootsGraph meaning
Δ > 0 and perfect squareTwo distinct rational rootsParabola crosses x-axis twice, nice integers/fractions
Δ > 0 non-squareTwo distinct irrational rootsCrosses twice; messy radicals
Δ = 0One real repeated rootParabola just touches x-axis (tangent) at vertex
Δ < 0No real roots (complex conjugates)Parabola entirely above or below x-axis
Real-life: In projectile motion, Δ > 0 means the object hits ground twice in time (launch and land times distinct ignoring start height), Δ = 0 means it just grazes, Δ < 0 means it never reaches ground level used in the equation.

8) Vieta’s Formulas — Relations Between Coefficients and Roots

If roots are x_1, x_2 (possibly complex), then for ax^2 + bx + c = 0:

  • Sum: x_1 + x_2 = -b/a
  • Product: x_1 x_2 = c/a

Why: Expand a(x - x_1)(x - x_2) = ax^2 - a(x_1 + x_2)x + a x_1 x_2 and match coefficients.

Use: Construct a quadratic with given roots or check answers quickly; also handy for sum/product questions without solving fully.

9) Quadratic Inequalities

To solve ax^2 + bx + c ▷ 0 (where ▷ is >, ≥, <, ≤):

  1. Solve the corresponding equation ax^2 + bx + c = 0 to find critical points.
  2. Draw a number line, mark the roots, and test intervals (consider the sign of a and opening direction).
  3. Use open circles for > or <; filled for ≥ or ≤.
Example: Solve x^2 - 5x + 6 ≤ 0.
Roots: 2, 3. Parabola opens up. ≤ 0 means “below or on” the x-axis → between/including the roots:
[2, 3].

10) Systems That Lead to Quadratics

  • Substitution: e.g., Solve the system y = 2x + 3 and y = x^2 - x - 1. Substitute to get a quadratic in x.
  • Geometry constraints: Rectangles with fixed perimeter or area conditions lead to quadratics.
  • Motion problems: Equal heights/time conditions often produce quadratics.

11) Optimization with Quadratics (Real-Life Applications)

A) Maximum Height (Projectile)

h(t) = -4.9 t^2 + v_0 t + h_0 (SI units). Vertex time t^* = -b/(2a) = v_0/(9.8). Height at vertex is the maximum.

B) Revenue & Profit

If demand is linear p(q) = m q + k, then revenue R(q) = q·p(q) is quadratic. Vertex gives the quantity for maximum revenue.

C) Fencing/Area

Given fixed perimeter, the area as a function of one side is a concave quadratic → maximize at vertex.

12) Strategy Guide — Which Method Should I Use?

  • Can you factor quickly? Try factoring first for speed.
  • No middle term (or easily isolated square)? Use square-root method.
  • Need vertex or max/min? Convert to vertex form by completing the square.
  • Numbers are messy or not factorable? Use the quadratic formula.
  • Need relationships (sum/product) without solving? Use Vieta.

13) Common Mistakes & Fixes

  • Forgetting the ± when taking square roots → always write both.
  • Sign errors in Δ = b^2 − 4ac → compute carefully, bracket negatives.
  • Dividing only part of the numerator by 2a → remember the entire -b ± √Δ is over 2a.
  • Dropping common factors before factoring → factor out GCF first to simplify.
  • For inequalities, mixing “inside” and “outside” intervals → sketch a quick parabola.

14) Long Exam-Style Worked Examples (Step-by-Step)

Example 1 — Full Mechanics (General → Vertex → Solve → Interpret)

Problem: A ball is launched from a 2 m platform with initial velocity 14 m/s upward. Height is h(t) = -4.9 t^2 + 14t + 2.

  1. Max height and when? Vertex time t^* = -b/(2a) = -14/(2·-4.9) = 14/9.8 ≈ 1.4286 s.
  2. Height at t^*: h(t^*) = -4.9(1.4286)^2 + 14(1.4286) + 2 ≈ -10 + 20 + 2 ≈ 12 m (compute precisely if required).
  3. Landing time (height 0): Solve -4.9 t^2 + 14 t + 2 = 0 using quadratic formula:
    a=-4.9, b=14, c=2, Δ = 14^2 - 4(-4.9)(2) = 196 + 39.2 = 235.2.
    t = [-14 ± √235.2]/(2·-4.9).
    Positive root: t ≈ [ -14 + 15.342 ] / -9.8 ≈ 1.342 / -9.8 ≈ -0.137 (reject), or
    t ≈ [ -14 - 15.342 ] / -9.8 ≈ (-29.342)/-9.8 ≈ 2.994 s.
    The ball lands ~ 2.99 s after launch.
  4. Interpretation: Parabola opens down (a<0). Vertex is the max height; zeros are launch height crossing (may be negative time extraneous) and landing time.
Exam tip: If a negative time appears, it’s just the “other” solution given by algebra, not physically meaningful.

Example 2 — Factoring via AC Method (Messy Coefficients)

Problem: Solve 6x^2 - 7x - 3 = 0.

  1. a·c = 6·(-3) = -18. Need two numbers multiplying to −18 and adding to −7 → −9 and +2.
  2. Split middle: 6x^2 - 9x + 2x - 3.
  3. Group: 3x(2x - 3) + 1(2x - 3) = (2x - 3)(3x + 1).
  4. Set each to zero: 2x - 3 = 0 ⇒ x = 3/2; 3x + 1 = 0 ⇒ x = -1/3.

Example 3 — Quadratic Inequality

Problem: Solve -x^2 + 4x + 5 > 0.

  1. Equation roots: -x^2 + 4x + 5 = 0 ⇒ x^2 - 4x - 5 = 0 → factors: (x-5)(x+1)=0 → roots -1, 5.
  2. Parabola opens down (a=-1), so it is above the axis between the roots.
  3. Strict inequality: (-1, 5).

Example 4 — Vieta Construction

Problem: Construct a quadratic with roots 3 and −7 and leading coefficient 2.

x_1 + x_2 = -4, x_1 x_2 = -21. So a=2, and standard form is 2[x^2 - (sum)x + (product)] = 2(x^2 + 4x - 21) → 2x^2 + 8x - 42 = 0.

Example 5 — Geometry Optimization (Longest Style)

Problem: A rectangular garden has perimeter 60 m. Find dimensions with maximum area.

  1. Let width be w, length ℓ. Perimeter: 2(w + ℓ) = 60 ⇒ ℓ = 30 - w.
  2. Area: A(w) = w(30 - w) = -w^2 + 30w (a quadratic opening down).
  3. Vertex at w^* = -b/(2a) = -30/(2·-1) = 15. Then ℓ = 30 - 15 = 15.
  4. Max area occurs for a square 15 m × 15 m; A = 225 m².
  5. Why: For fixed perimeter, the square maximizes area; the vertex gives the optimum.

15) Practice Set with Answers (Concise)

  1. Solve: x^2 - 4x - 12 = 0. x=6, -2 (factor).
  2. Solve: 3x^2 + 2x - 7 = 0. x = [-2 ± √(4 + 84)]/6 = [-2 ± √88]/6 = (-1 ± √22)/3.
  3. Discriminant of 5x^2 + 10x + 6 = 0. Δ=100-120=-20 <0 → complex roots.
  4. Vertex of y = 2x^2 - 8x + 1. h=8/4=2; k=2(4)-16+1=8-16+1=-7 ⇒ (2, -7).
  5. Roots sum/product for 7x^2 - 3x - 2=0. Sum=3/7, Product=-2/7.
  6. Solve inequality x^2 - 9x + 14 ≥ 0. Roots 2,7; opens up → outside intervals → (-∞,2] ∪ [7,∞).
  7. Write quadratic with roots 4 and -1, leading coefficient 3. 3(x-4)(x+1)=3x^2 - 9x - 12.
  8. Time to hit ground for h(t) = -5t^2 + 5t + 10. Solve -5t^2 + 5t + 10=0 → t = [ -5 ± √(25 + 200)]/(-10) = [-5 ± 15]/(-10) ⇒ t=-1, 2 → physical 2 s.
  9. Complete the square: x^2 + 10x + 7. (x+5)^2 - 25 + 7 = (x+5)^2 - 18.
  10. Minimum value of y = x^2 - 12x + 41. Vertex at x=6, y_min = 36 - 72 + 41 = 5.

16) Quick Reference Sheet (Memorize These)

  • Standard: y=ax^2+bx+c
  • Vertex: y=a(x-h)^2+k, h=-b/(2a), k=f(h)
  • Factored: y=a(x-r_1)(x-r_2)
  • Axis: x=-b/(2a)
  • Quadratic Formula: x = [-b ± √(b^2 - 4ac)]/(2a)
  • Discriminant: Δ=b^2-4ac (sign tells root type)
  • Vieta: x_1 + x_2 = -b/a, x_1x_2 = c/a
  • Completing square add-in: (\tfrac{b}{2a})^2 (after dividing by a)
  • Optimization: vertex gives max/min; for a<0 → max, a>0 → min.
How to think (fast mental checklist):
  1. Is it really quadratic (power 2)?
  2. Can I factor? If not, compute Δ.
  3. Need vertex or max/min? Go to vertex form.
  4. Need relationships only? Use Vieta.
  5. For inequalities, sketch parabola and read intervals.

Extra Real-Life Mini-Examples (each ties a concept)

  • Break-even (business): Revenue R(q)=-0.2q^2+40q, cost C(q)=12q+50. Profit P=R-C=-0.2q^2+28q-50. Zeros of P (quadratic formula) are break-even outputs.
  • Safety stopping distance: d(v)=αv+βv^2 (reaction + braking). Given target max d, solve quadratic in v to find safe speed.
  • Design arch: Parabolic arch y=a(x-h)^2+k; use vertex for highest point and zeros for supports.

Self-Check (Mini Quiz)

  1. Given y = -3(x-4)^2 + 7, state opening, vertex, axis, max/min and value.
    Ans: Opens down; vertex (4,7); axis x=4; maximum 7 at x=4.
  2. For 4x^2 + 4x + 1 = 0, compute Δ and roots.
    Ans: Δ=16-16=0; repeated root x=-b/(2a) = -4/8 = -1/2.
  3. Make a quadratic with sum of roots 8 and product 15.
    Ans: monic: x^2 - 8x + 15 = 0.

With these notes, you can identify, graph, solve, analyze, and apply quadratics in real contexts — and explain why every step works.

Reference Book: N/A

Author name: SIR H.A.Mwala Work email: biasharaboraofficials@gmail.com
#MWALA_LEARN Powered by MwalaJS #https://mwalajs.biasharabora.com
#https://educenter.biasharabora.com

:: 1.1::

⬅ ➡