Study guide: Factorising and Solving Quadratics

📋 Course Outline

1. Common Factors and Grouping
2. Factoring Quadratic Expressions
3. Special Products and Cubics
4. Quadratic Equations by Factoring
5. The Quadratic Formula
6. Completing the Square
7. Quadratic Models and Applications
8. Linear–Nonlinear Systems
9. Mathematicians and Applications

📖 1. Common Factors and Grouping

🔑 Key Concepts & Definitions

• Factorising: The reverse of expanding, writing an expression as a product of factors by taking out common factors.

★ Must-know

📌 A factorisation is complete only when all common factors have been taken outside the brackets.

• 🔄 Factorise by grouping in these steps:
  1. Pair terms so each pair has a common factor
  2. Factor each pair
  3. Take out the common bracket

Further detail

• The expression $ah+ak+bh+bk$ factorises by grouping as $(h+k)(a+b)$.

📖 2. Factoring Quadratic Expressions

★ Must-know

• For a monic quadratic $x^2+bx+c$, find two numbers whose product is $c$ and whose sum is $b$, then use them as the constants in the two factors.

• For $ax^2+bx+c$ with $a\ne1$, one method seeks two numbers with product $ac$ and sum $b$, splits the middle term using them, groups the terms, and factors out the common bracket.

Further detail

• The quadratic $x^2+6x+8$ factorises as $(x+2)(x+4)$ because 2 and 4 multiply to 8 and add to 6.

• For a quadratic with leading coefficient 2, inspection starts with factors whose first terms are $2x$ and $x$.

📖 3. Special Products and Cubics

★ Must-know

📌 A difference of two perfect squares factors as $A^2-B^2=(A-B)(A+B)$.

• A cubic expression of the form $ax^3+bx^2+cx$ can be factorised by taking out the common factor of $x$ (or a multiple of $x$) and then factorising the remaining quadratic.

Further detail

• The expression $4a^2-b^2$ factorises as $(2a-b)(2a+b)$.

💡 Memory Hook

A difference of squares factors as a pair with opposite signs: (A − B)(A + B).

📖 4. Quadratic Equations by Factoring

🔑 Key Concepts & Definitions

• Quadratic equation: An equation of the form $ax^2+bx+c=0$, where $a\ne0$; the coefficients $b$ and $c$ may be zero, and no power of the variable may exceed 2.

★ Must-know

• A quadratic equation can have zero, one, or two solutions.

📌 If a product of factors equals zero, at least one factor must equal zero; set each factor equal to zero to find the solutions.

• 🔄 Solve a factorisable quadratic in these steps:
  1. Rearrange the equation to equal zero
  2. Factorise the quadratic
  3. Set each factor equal to zero
  4. Solve the resulting equations

Further detail

• The equation $x^2-25=0$ has solutions $x=5$ and $x=-5$.

📖 5. The Quadratic Formula

★ Must-know

📐 Formula - For $ax^2+bx+c=0$, the solutions are $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.

Further detail

📌 Use the quadratic formula when the equation cannot be solved by factorisation; if the expression under the square root is a square number, factorisation was possible.

• When a quadratic-formula answer must be rounded, retain exact x-values while calculating dependent values and round only at the end.

• For $2x^2-3x-4=0$, the exact solutions are $x=\frac{3+\sqrt{41}}{4}$ and $x=\frac{3-\sqrt{41}}{4}$.

📖 6. Completing the Square

★ Must-know

• For a monic quadratic, complete the square by replacing $x^2+bx$ with $(x+\frac b2)^2-(\frac b2)^2$, then isolate the square and take both square roots.

• Because a square is nonnegative, $(x-h)^2+k$ has minimum value $k$ when $x=h$.

Further detail

• The equation $x^2+6x+10=0$ has no real solutions because completing the square gives a square equal to a negative number.

• Completing the square also works when the coefficient of $x^2$ is not 1 by first factoring that coefficient from the quadratic and linear terms.

📖 7. Quadratic Models and Applications

★ Must-know

• 🔄 Model and solve a quadratic word problem in these steps:
  1. Define a variable
  2. Express the relationships algebraically
  3. Form and solve the quadratic equation
  4. Reject roots that violate the context

📌 In a geometric or physical problem, a mathematically valid root must be rejected if it gives an impossible length or other quantity in the context.

Further detail

• If two positive numbers differ by 3 and have product 88, their values are 8 and 11.

• The area of a rectangle is the product of its length and width.

📖 8. Linear–Nonlinear Systems

★ Must-know

📌 When one equation in a simultaneous pair is linear and the other is nonlinear, use substitution rather than the elimination method used for two linear equations.

• 🔄 Solve the simultaneous pair in these steps:
  1. Rearrange the linear equation if needed
  2. Substitute into the nonlinear equation
  3. Solve the resulting quadratic
  4. Substitute each x-value back to find its paired y-value

Further detail

📌 When a quadratic formula is needed in a simultaneous-equation problem, use exact x-values to find the corresponding y-values and round both coordinates only at the end.

💡 Memory Hook

Substitute, solve the quadratic, then substitute back.

📖 9. Mathematicians and Applications

📝 Essential Points

• Srinivasa Ramanujan (1887−1920) was a self-taught Indian mathematician who contributed to analysis and number theory and solved problems then considered unsolvable.

• In 1914, Srinivasa Ramanujan travelled to England to work with G. H. Hardy at Trinity College in Cambridge.

• In 1918, Srinivasa Ramanujan became a Fellow of the Royal Society.

• Ingrid Daubechies (1954−) is a Belgian physicist and mathematician best known for her work on image compression technology.

• Ingrid Daubechies’s image-processing methods have helped scientists extract information from bone and tooth samples and establish the age and authenticity of artworks, including paintings by Vincent van Gogh and Rembrandt.