Pure

• What is the procedure for expanding brackets in algebra?

  Multiply each term in the first expression by each term in the second expression and simplify by collecting like terms.
• How do you simplify the expression \((x + 2)(x + 3)\)?

  By expanding it to \(x^2 + 3x + 2x + 6\) and then collecting like terms to get \(x^2 + 5x + 6\).
• What is the expanded form of \(3(p + 3)(p + 2)\)?

  It expands to \(3p^2 + 15p + 18\).
• What is the first step in expanding \((q + 1)(q + 2)(q + 3)\)?


  Start by expanding the first two brackets \((q + 1)(q + 2)\).
• What do you do after expanding the first two brackets in \((q + 1)(q + 2)(q + 3)\)?

  Rewrite the expression as \((q^2 + 3q + 2)(q + 3)\) and then expand it.
• What is the definition of factorising in algebra?

  Factorising is the reverse of expanding brackets, finding the factors of a given expression.
• How do you factorise a quadratic expression of the form \(ax^2 + bx + c\)?

  Calculate the product \(a \times c\) and find two factors that add up to \(b\).
• What is the first step in factorising the expression \(x^2 + 5x + 6\)?

  Identify \(a = 1\), \(b = 5\), and \(c = 6\) and find factors of \(a \times c\) that add up to \(b\).
• What are the two numbers that satisfy the conditions for factorising \(x^2 + 5x + 6\)?

  The two numbers are \(3\) and \(2\).
• What is the difference of two squares formula?

  The formula is \(x^2 - y^2 = (x + y)(x - y)\).
• What are surds in mathematics?

  Surds are irrational numbers that are expressed in the exact form of \(\sqrt{a}\) where \(a\) is not a square number.
• How do you rationalise a denominator that contains a surd?

  Multiply the numerator and denominator by the conjugate of the denominator.
• What is the first step in simplifying the expression \(\frac{1}{5 + \sqrt{44}}\)?


  Rewrite it as \(\frac{1}{5 + \sqrt{4 \times 11}}\).
• What is the final simplified form of \(\frac{1}{5 + \sqrt{44}}\) after rationalising the denominator?

  The final simplified form is \(-\frac{5 - 2\sqrt{11}}{19}\).
• What are the four key index laws in mathematics?

  1. \(a^m \times a^n = a^{m+n}\)
  2. \(a^m \div a^n = a^{m-n}\)
  3. \((a^m)^n = a^{mn}\)
  4. \((ab)^m = a^m b^m\)
• How do you simplify the expression \(x^2 \times x^3\) using index laws?

  It simplifies to \(x^{2+3} = x^5\).
• What is the result of \(\frac{y^2}{y^3}\) using index laws?

  It simplifies to \(y^{2-3} = y^{-1}\).
• What is the simplified form of \((z^4)^2\)?

  It simplifies to \(z^{4 \times 2} = z^8\).
• How do you simplify the expression \((x^2y^3)^3\)?


  It simplifies to \(x^{2 \times 3}y^{3 \times 3} = x^6y^9\).
• What is the rule for negative indices?

  For a negative index, \(a^{-n} = \frac{1}{a^n}\).
• How do you express \(a^{-\frac{1}{2}}\) in terms of a square root?

  It can be expressed as \(\frac{1}{\sqrt{a}}\).
• What is the simplified form of \(a^{\frac{1}{2}}\)?


  It is equal to \(\sqrt{a}\).
• How do you express \(s^{\frac{1}{2}}\) in terms of \(t\) if \(s = t^3\)?

  It can be expressed as \(t^{\frac{3}{2}}\).
• What is the process for factorising the expression \(x^2 + 5x + 6\)?

  Identify \(a\), \(b\), and \(c\), find factors of \(a \times c\) that add up to \(b\), and rewrite the expression.
• What are the two factors of \(x^2 + 5x + 6\)?

  The two factors are \(3\) and \(2\).
• How do you factorise \(x^2 + 5x + 6\) after finding the factors \(3\) and \(2\)?


  Rewrite it as \(x^2 + 2x + 3x + 6\) and then factor by grouping.
• What is the final factorised form of \(x^2 + 5x + 6\)?


  The final factorised form is \((x + 2)(x + 3)\).
• What are the rules for simplifying surds?


  • \(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\)
  • \(\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}\)
• How do you simplify \(\sqrt{36x^2} \div \sqrt{6}\)?

  It simplifies to \(\sqrt{6x^2} = x\sqrt{6}\).
• What is the process for rationalising the denominator of a fraction with a surd?

  Multiply the numerator and denominator by the conjugate of the denominator.
• What is the final result of rationalising the denominator in the expression \(\frac{1}{5 + \sqrt{44}}\)?

  The final result is \(-\frac{5 - 2\sqrt{11}}{19}\).
• What is the standard form of a quadratic equation?


  It is represented as $ax^2 +$$bx +$$c =$$0$, where $a \neq 0$.
• What must be done to solve a quadratic equation?

  The equation must be rewritten in the form $ax^2 +$$bx +$$c =$$0$.
• How do you solve for 'x' after factorizing a quadratic equation?

  You set each factor equal to zero and solve for 'x'.
• How many real solutions can a quadratic equation have?

  A quadratic equation can have one, two, or no real solutions.
• What is the first step in solving the quadratic equation $3x^2 - 2x - 8 =$$0$?


  The first step is to factor the equation.
• What is the quadratic formula used for?

  It is used to solve quadratic equations that may not be easily factorable.
• What is the quadratic formula?

  The quadratic formula is $x =$$\frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$.
• What does completing the square involve?

  It involves rewriting equations or expressions in a specific form to simplify solving.
• How can the expression $x^2 +$$bx +$$c$ be rewritten using completing the square?


  It can be rewritten as $\left(x + \frac{b}{2}\right)^2 - \left(\frac{b}{2}\right)^2 +$$c$.