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12. QUADRATIC EQUATIONS

 

There are three ways to solve quadratic equations:

(i)                  factorising

(ii)                completing the square

(iii)               by substituting in the formula;  x =

 

To illustrate this we will solve the equation x2 + 2x -8 = 0 by each of the three methods.

(i) factorising:    x2 + 2x -8 = 0 can be factorised to (x – 2)( x + 4) = 0

                        Hence (x – 2) = 0 giving x = 2 OR (x + 4) = 0 giving x = -4

                        x = 2 or -4

 

(ii) completing the square: Complete the square by adding the square of half the coefficient of x.

x2 + 2x -8 = 0  The coefficient of x is 2. Half the coefficient of x is 1. So we add 1 and subtract 1.

x2 + 2x  +1 -1 -8 = 0    x2 + 2x +1 -9 = 0         (x + 1)2 – 9 = 0            (x + 1)2 = 9

taking the square root of each side: (x + 1) =  3

(x + 1) = 3 gives x = 2 (x + 1) = -3 gives x = -4

x = 2 or -4.

 

(iii)               The formula; x =              x2 + 2x – 8 = 0

                  In this case a = 1, b = 2, c = -8

                  Substituting:      x =

 

                                          x =        =

                                          x =      = 2       OR       x =  = -4

                                          x = 2 or -4

 

Questions:

Solve the following quadratic equations by each of the above methods.

(i) x2 -2x – 8 = 0                (ii) x2 + 6x -16  = 0                  (iii) x2 + 6x + 8 = 0      

(iv) x2 -10x + 24 = 0          (v) x2 -12x + 36 = 0                 (vi) x2 – 18x + 80 = 0

(vii) x2 – x – 2 = 0              (viii) x2 + 3x + 2 = 0                 (ix) x2 + x – 6 = 0

(x) x2 + x – 12 = 0              (xi) x2 – 4x – 12 = 0                 (xii) x2 + x – 20 = 0

(xiii) g2 -8g + 12 = 0           (xiv) m2 + 2m – 15 = 0 (xv) 2k2 + 7k + 3 = 0

Answers:

(i) x = 4 or -2                     (ii) x = 2 or -8                          (iii) x = -2 or -4           

(iv) x = 4 or 6                     (v) x = 6                                   (vi) x = 8 or 10

(vii) x = 2 or -1                   (viii) x = -1 or -2                      (ix) x = 2 or -3

(x) x = 3 or -4                    (xi) x = 6 or -2                         (xii) x = 4 or -5

(xiii) g = 2 or 6                   (xiv) m = 3 or -5                       (xv) k = - ½ or -3

 

MORE QUADRATIC EQUATIONS

 

  1. Use factorisation to solve the following quadratic equations:

(i) x2 – 4x – 12 = 0       (ii) x2 + 7x + 12 = 0      (iii) x2 – 11x + 24 = 0

(iv) x2 + 3x - 40 = 0     (v) x2 - 10x + 16 = 0    (vi) 2x2 – x – 15 = 0

(vii) 3x2 + x – 10 = 0    (viii) 4x2 – 4x – 3 = 0    (ix) 9x2 - 6x - 8 = 0

(x) 6x2 – 13x + 6 = 0

2.   Use the formula x =  to solve the following quadratic equations. Give your answers to 1 decimal point.

      (i) x2 + x - 4 = 0           (ii) x2 + 12x + 3 = 0      (iii) x2 – 2x - 5 = 0

      (iv) 2x2 –10 x + 1 = 0   (v) 5x2 – 2x - 3 = 0      (vi) 2x2 + x -1 = 0

      (vii) 3x2 + 6x -2 = 0      (viii) 6x2 + 4x -3 = 0     (ix) 4x2 – x – 7 = 0

      (x) 3x2 + 4x – 9 = 0

3.   Use the formula from Q.2. to show that the equation x2 + x + 4 = 0 has no solution.

      Draw the graph y = x2 + x + 4 and state why the graph shows that the equation x2 + x + 4 = 0 has no solution.

4.   Draw the graph:  y = x2 – 2x – 3

Use your graph to solve the equations x2 – 2x – 3 = 0 and x2 – 2x – 8 = 0.

Answers:

1.   (i) x = 6 or -2   (ii) x = -3 or -4  (iii) x = 3 or 8  (iv) x = 5 or -8  (v) x = 2 or 8

      (vi) x = 3 or -2.5  (vii) x = 5/3 or -2  (viii) x = 1.5 or – 0.5  (ix) x = 4/3 or -2/3

      (x) x = 1½ or 2/3

2.   (i) x = 1.6 or -2.6  (ii) x = -0.3 or -11.7  (iii) x = 1.4 or 3.4  (iv) x = 0.1 or 4.9

      (v) x = 1 or -0.6  (vi) x = ½ or -1  (vii) x = 0.3 or -2.3  (viii) x = 0.4 or –1.1 (ix) x = 1.5 or -1.2  (x) x = 1.2 or – 2.5

3.   x =  =               graph:

      Since  has no solution and the graph does not cut the x-axis then the equation x2 + x + 4 = 0 has no solution.

4.   The solutions to x2 – 2x – 3 = 0  are the points where y = 0 i.e. x = -1 or 3.

                                          Graph:

      The solution to x2 – 2x – 8 = 0 is the same as x2 – 2x – 3 = 5 i.e. the intersection of the graph y = x2 – 2x – 3 and the line y = 5 i.e. x = 4 or -2

 

QUADRATIC FUNCTIONS

 

A quadratic function is one of the form y = ax2 + bx + c where a, b &c are constants.

A graph of this function is a parabola.

If a is positive (a>0) the graph is concave up.

If a is negative (a<0) the graph is concave down.

If a = 0 the function is not quadratic and the graph is not a parabola.

 

The graph of a parabola may cut the x-axis at two points, touch the x-axis at one point only or lie completely above or below the x-axis.

The x values where the function touches or cuts the x-axis are known as the roots of the function.