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Unit 3 Quadratic Equations - Coggle Diagram
Unit 3 Quadratic Equations
3.1 Concept of quadratic equations
A quadratic equation is an equation where the highest exponent of the variable (usually "x") is a square
Quadratic equation is an equation that can be written in standard form
standard form: ax2 + bx + c = 0 (a ≠ 0)
quadratic term + linear term + constant = 0
The root/solution of a quadratic equation is the set of all values that, when substituted for unknown, make the equation true
3.2 Solving quadratic equations
3.2.1 Factoring
Zero-product property: If AB = 0, then A = 0 or B = 0
Step 1: Write the equation in standard form
Step 2: Factor the quadratic expression
Step 3: Use the Zero Product Property to set each factor to 0
Step 4: Solve each equation
3.2.2 Find the square roots
Given a quadratic equation ax2 + c = 0 (a ≠ 0) can be written as x2 = -(c/a); Let -(c/a) = d, x2 = d
d > 0, the equation has two distinct real roots: x1 = √d, x2 = -√d
d = 0, the equation has two identical real roots: x1 = x2 = 0
d < 0, the equation has no real roots
3.2.3 Completing the square
Solving the equation ax2 + bx + c = 0 by completing the square
If a ≠ 1, divide everything on both sides by a
isolate the constant on the right side of the equation
Add (1/2b)2 to both sides
Factor the now perfect square on the left side
Use the square root property to complete the solution
3.2.4 Quadratic formula
Step 1: Write the equation in standard form
Step 2: Identify the value of a, b, c in standard form of the equation
Step 3: Evaluate b2 - 4ac
Step 4: If b2 - 4ac ≥ 0, substitute a, b, c into the quadratic formula; if b2 - 4ac < 0, there is no real roots
If b2 - 4ac > 0, x = (-b ± √b2 - 4ac) / 2a
If b2 - 4ac = 0, x1 = x2 = -(b / 2a)
3.3 Discriminant of quadratic equations
In the quadratic formula, b2 - 4ac, which is under the square root sign, is called the discriminant
The symbol delta ∆ is used to represent the discriminant, so ∆ = b2 - 4ac
The quadratic formula becomes x = (-b ± √∆) / 2a
To use the discriminant to determine the nature of the roots of an equation: 1. write the standard form 2. Identify the values of a, b, c 3. evaluate ∆ 4. write the conclusion
To determine the conditions based on the number of roots
Identify the values of a, b, c
Evaluate ∆
Construct an inequality or an equation based on the number of roots
Solve the inequality or equation
3.4 Vieta's Formula
3.4.1 Vieta Formula for Quadratic equation
x1 + x2 = -(b/a)
x1· x2 = c/a
3.4.2 Application of Vieta's Formula
Determining the value of an undetermined variable by using the relationship between roots and coefficients
Using vieta formula to simplify: x1 + x2, x1·x2
Evaluate the transformation of formulas (Don't forget to check the discriminant)
3.5 Solving problems with quadratic equations
3.5.1 Factoring by quadratic formula
Factoring ax2 + bx + c = 0 (a ≠ 0)
Let the quadratic trinomial equal to 0.
Evaluate b2 - 4ac
Solve the equation
Factoring by the formula
3.5.2 Solving word problem by quadratic equations
A construction team plans to use an existing 80-meter-long wall and a 120-meterlong metal fence to enclose a rectangular temporary storage area for construction
waste. The fence will only be used for the three sides not covered by the wall
Example question
Is it possible to enclose a rectangular storage area that meets the given conditions?
If yes, determine the lengths of the two adjacent sides of the rectangle.
(1) Area of the rectangle is 2000 m2
(2) Area of the rectangle is 1800 m2
(3) Area of the rectangle is 1152 m2
General Steps
Equivalence
Variable
Equation
Solve
Check
Ans