This equation is a consequence of Pythagoras theorem and is illustrated below.
The equation of a circle is a quadratic equation because of theandterms.
]]>We might multiply this out to obtainNow we equateto this and solve for the coefficients a, b, c.
Example: Complete the square for the expression
Equating coefficients of
Equating coefficients of
Equating constant terms:
Hencein completed square form is
Having completed the square we can solve the equation
Example: Complete the square for the expressionHence solve the equation
Equating coefficients of
Equating coefficients of
Equating constant terms:
Hencein completed square form is
Now we can solve
]]>Starting with a horizontal plane the intersction describes acircle with equation
Tilting the plane with respect to the cone deforms the circle intoan ellipse with equation
Tilting it still further so that the plane is parallel to aslanted side of the cone means that the ellipse never closes at thelower end, andf it becomes a parabola with equation
Ttilting the plane further to the vertical so that it passesthrough the vertex where the upper and lower cones meet divides theintersection into two parts, with one each intersecting the upper andlower halves. The curve is now a hyperbola with equation
The constants for each equation are not connected in general.
]]>Starting fromdivide through by a to give(2) then write
Substitute this into (2) to give
Now make x the subject. Start by moving the last two terms to the right hand side.
Collect the terms on the right by adding them as fractions.
Square root both sides to give
Subtraction ofgives
]]>Example: The graph below passes through the p[ointsand
Substitute these points into the equation (1) to give the simultaneous equations
(2)
(3)
(4)
Substitutinginto (3) and (4) gives
(5)
(6)
(6)-2*(5) gives
From (5) then
]]>For example, ifthenand (note we always take the positive square root) so
Sometimes we need to factorise with any common factors before we can use the difference of squares factorisation.
Now putso thatand
The roots of the difference of squaresis the set of values offor whichThe roots are always equal in magnitude and opposite in sign. If an quadratic expression factorises into a difference of squares this is always the case. If we sketch this function, it is a quadratic graph with the line of symmetryand minimum at
The roots of the difference of squaresare x=-a and x=a. If we sketch this function, it is a quadratic graph with the line of symmetryand minimum atThis is a 'sad' curve, as opposed to the 'happy' curve above.
The above examples are quadratics, withterms but this need not be the case.
]]>Factorise
Take out any common factor. Every term in the above expression has a factor 3, so we may write the expression as
Multiply the coefficient ofby the constant term:Find the two factors of this product which add to give the coefficient ofwhich in this case is -1.
Rewrite the term in brackets using these two factors:
Take out common factors for each pair:
Factorise completely:
Example
Factorise
Take out common factors:
Multiply the coefficient ofby the constant term:Find the two factors of this product which add to give the coefficient ofwhich in this case is -7: -1 and -6.
Rewrite the term in brackets using these two factors:
Take out common factors for each pair:
Factorise completely:
We can findif we have the coordinates of some point on the curve.
Example: A quadratic equation has roots 1, 3 and passes through the pointFind the equation of the curve.
Because the roots are 1 and 3, the equation of the curve must be of the formSincelies on the curve, we must have
The equation of the quadratic is
Example: A quadratic equation has the single root 2 and passes through the pointFind the equation of the curve.
Because the root is 2 the equation of the curve must be of the formSincelies on the curve, we must have
The equation of the quadratic is
Example: A quadratic equation has the rootsandand passes through the pointFind the equation of the curve.
Because the roots areandthe equation of the curve must be of the formon expanding tge brackets. Sincelies on the curve, we must have
The equation of the quadratic is
]]>Ifthe minimum will be wheresoand the minimum is at
Ifthe maximum will be wheresoand the maximum is at
For example, to find the minimum ofcomplete the square to getthen the minimum is at
To find the maximum ofcomplete the square to get then the maximum is at
]]>