Solving quadratic equations
- If xy=0, then x=0 or y=0.
- This property is sometimes called the zero product property
- The property can be expanded to:
- If (x-a)(x-b)=0, then x-a=0, or x-b=0
- To solve an equation by completing the square, take half the coefficient of x, square it, and add the result to both sides of the equation. This step creates a perfect square trinomial on the left side of the equation.
- In order to complete the square, the coefficient of the x^2 term must be 1. If the x^2 term has a coefficient other than 1, you can factor out the coefficient or divide through by the coefficient.
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The quadratic formula
- For any equation in the form ax^2+bx+c,
This equation is given in the IB formula booklet, but it can be useful to memorize.
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Roots of quadratic equations
- For a quadratic equation (-b±√(b^2-4ac))/2a=0
- If b^2 - 4ac>0, the equation will have two different real roots
- If b^2 - 4ac =0, the equation will have two equal real roots
- If b^2 - 4ac <0, the equation will have no real roots
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Graphs of quadratic equations
- For quadratic functions in the standard form y=ax^2+bx+c=0, the graph will cross the y axis at (0,c)
- The equation for the axis of symmetry is x=-b/2a
- When the basic function y=x^2 undergoes a transformation, the resulting functions can be written as y=a(x-h)^2 +k
- For quadratic functions in the form y=a(x-h)^2 +k, the graph will have its vertex at (h,k)
- For quadratic function in the form y=a(x-p)(x-q), the graph crosses the x axis at (p,0) and at (q,0)
- For quadratic function in the form y=a(x-p)(x-q), the axis of symmetry will have the equation x=(p+q)/2
- When the equation is in the form f(x)=a(x-h)^2+k, also known as the turning point form, the vertex will be (h,k).
- When the equation is written in factorized form f(x)=a(x-p)(x-q), the graph will cross the x axis at (p,0) and at (q,0)
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