Solving quadratic equations

• What is the first step in solving a quadratic equation using factorisation?


  Rearrange it into the form \( ax^2 + bx + c = 0 \)

  
• Why is it easier to use the side where \( a \) is positive when solving a quadratic equation?

  It simplifies the factorisation process

  
• If you have the equation \( (x + 4)(x - 1) = 0 \), what are the two equations you need to solve?

  x + 4 = 0 or x - 1 = 0

  
• What does it mean if \( A \times B = 0 \)?

  Either \( A = 0 \) or \( B = 0 \)

  
• How do you solve the equation \( (x - 3)(x + 7) = 0 \)?


  Set \( x - 3 = 0 \) and \( x + 7 = 0 \)

  
• What are the two solutions for the equation \( (x - 3)(x + 7) = 0 \)?

  x = 3 or x = -7

  
• What is a common mistake when solving equations like \( x(x - 4) = 0 \)?

  Dividing both sides by \( x \) at the beginning

  
• How do you solve \( 2x - 3(3x + 5) = 0 \)?

  Set \( 2x - 3 = 0 \) and \( 3x + 5 = 0 \)

  
• What are the two solutions for the equation \( 2x - 3(3x + 5) = 0 \)?

  x = \frac{3}{2} or x = -\frac{5}{3}

  
• What is the first step in solving a quadratic equation by completing the square?


  Replace the first two terms \( x^2 + bx \) with \( (x + p)^2 - p \)

  
• How do you express \( x^2 + 10x + 9 = 0 \) in completed square form?


  (x + 5)^2 - 16 = 0

  
• How does the quadratic formula relate to completing the square?

  The quadratic formula is derived from completing the square

  
• What is the discriminant in the quadratic formula?

  The part of the formula under the square root, \( b^2 - 4ac \)

  
• What does it indicate if the discriminant \( b^2 - 4ac > 0 \)?

  There are 2 different solutions

  
• What does it indicate if the discriminant \( b^2 - 4ac = 0 \)?

  There is only 1 solution, sometimes called "two repeated solutions"

  
• What does it indicate if the discriminant \( b^2 - 4ac < 0 \)?

  There are no solutions

  
• How do you use a calculator to solve quadratic equations?

  Type the equation into the calculator and use the quadratic formula

  
• What are the steps to solve a quadratic equation by factorising?


  1. Rearrange into the form \( ax^2 + bx + c = 0 \)
  2. Factorise the quadratic
  3. Set each factor equal to zero
  4. Solve for \( x \)

  
• What are the steps to solve a quadratic equation by completing the square?


  1. Rearrange into the form \( x^2 + bx + c = 0 \)
  2. Replace \( x^2 + bx \) with \( (x + p)^2 - p \)
  3. Rearrange to make \( x \) the subject
  4. Solve for \( x \) using \( \pm \sqrt{} \)

  
• What is the quadratic formula?

  • The quadratic formula is \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)
  • Used to find solutions of \( ax^2 + bx + c = 0 \)

  
• What does the discriminant tell you about a quadratic equation?

  • If \( b^2 - 4ac > 0 \): 2 different solutions
  • If \( b^2 - 4ac = 0 \): 1 solution (two repeated solutions)
  • If \( b^2 - 4ac < 0 \): no solutions

  
• What is the formula for the discriminant in a quadratic equation?

  The discriminant is given by \( b^2 - 4ac \).

  
• What does the sign of the discriminant indicate about the solutions of a quadratic equation?

  The sign indicates whether there are 0, 1, or 2 solutions.

  
• What does it mean if \( b^2 - 4ac > 0 \)?

  It means there are 2 different solutions.

  
• What does it mean if \( b^2 - 4ac = 0 \)?

  It means there is only 1 solution, sometimes called "two repeated solutions".

  
• What does it mean if \( b^2 - 4ac < 0 \)?

  It means there are no solutions.

  
• What are complex solutions in the context of quadratic equations?

  Complex solutions are those that include \( i \) terms.

  
• What does it indicate if \( b^2 - 4ac \) is a perfect square number?

  It indicates that the quadratic expression could have been factorised.

  
• Can you use a calculator to solve quadratic equations?

  Yes, but a method must still be shown.

  
• What must be done to a quadratic equation before using a calculator to solve it?

  The quadratic equation must have "= 0" on the right-hand side.

  
• How should final answers be presented when solving quadratic equations?

  Final answers should be presented as specified, for example, correct to 2 decimal places.

  
• What is the procedure to solve the equation \( 3x^2 - 2x - 4 = 0 \) using the quadratic formula?

  1. Identify \( a = 3, b = -2, c = -4 \).
  2. Substitute into the quadratic formula.
  3. Use a calculator to find solutions:

  • First solution: \( x = 1.54 \)
  • Second solution: \( x = 0.869 \)

  1. Present both answers together.

  
• When should you solve a quadratic equation by factorisation?

  • When explicitly asked to solve by factorisation.
  • When solving two-term quadratic equations.
  • When the quadratic expression can be easily factorised.

  
• When should you use the quadratic formula to solve a quadratic equation?

  • When asked to leave solutions correct to a given accuracy.
  • When the quadratic formula may be faster than factorising.
  • When in doubt, as it always works.

  
• When should you solve a quadratic equation by completing the square?


  • When instructed to complete the square in part (a) of a question.
  • When making \( x \) the subject of harder formulas.

  
• What should you do if your calculator gives whole numbers or fractions as solutions?

  This means the quadratic equation does factorise.

  
• What is the process to solve \( x^2 - 7x + 2 = 0 \) using the quadratic formula?

  1. Substitute \( a = 1, b = -7, c = 2 \) into the formula.
  2. Use a calculator to find solutions:

  • First solution: \( x = 6.70 \)
  • Second solution: \( x = 0.30 \)

  1. Round final answers to 2 decimal places.

  
• What is the method to solve \( 16x^2 - 82x + 45 = 0 \) if factorisation is not obvious?

  1. Use the quadratic formula with \( a = 16, b = -82, c = 45 \).
  2. If factorisation is spotted, use that method instead.

  
• How do you complete the square for \( x^2 + 6x + 5 = 0 \)?


  1. Find \( p \) by halving the middle number.
  2. Write \( x + 6x \) as \( (x + 3)^2 - 9 \).
  3. Solve the equation by taking square roots and isolating \( x \).

  
• What are the final solutions for \( x^2 + 6x + 5 = 0 \) after completing the square?

  The solutions are \( x = -5 \) and \( x = -1 \).