Chapter 15: Completing the Square and Simultaneous Equations

Completing the square

• If a quadratic expression is written in the form (x+p)^2 + q it is in completed square form
• You can solve quadratic equations which don’t have integer answers by completing the square

Useful identities

• If you learn these two identities you can save time when you are completing the square
• x^2 + 2bx + c = (x+b)^2 - b^2 + c
• x^2 - 2bx + c = (x-b)^2 -b^2 + c

Positive and negative roots

• Remember that any positive number has two square roots, one positive and one negative
• If you ‘square root’ both sides of an equation you need to use +- to show that there are two square roots

Simultaneous equations

• Simultaneous equations have two unknowns
• You need to find the values for the two unknowns which make both equations true

Algebraic solution

• Number each equation
• If necessary, multiply the equations so that the coefficients of one unknown are the same
• Add or subtract the equations to eliminate that unknown
• Once one unknown is found use substitution to find the other
• Check the answer by substituting both values into the other equation

Easier eliminations

• You can save time by choosing the right unknown to eliminate
• Look for one of these
  • If an unknown appears on its own in one equation you only need to multiply one equation to eliminate that unknown
  • If an unknown has different signs in the two equations tou can eliminate by adding

Check it

• Always use the equation you didn’t substitute into to check your answer

Graphical solution

• You can solve these simultaneous equations by drawing a graph
• The coordinates of the point of intersection give the solution to the simultaneous equation