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EQUATIONS IN THE QUADRATIC FORM 142. A complete quadratic equation contains the second and first power of the unknown number, that is, one exponent is twice the other. Hence, any equation that contains only two powers of the unknown number is said to be in the quadratic form, if one of the exponents is twice the other. Such equations may be solved by the methods used in solving quadratics.

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GIROLAMO CARDANO

(From an Old Engraving) Born at Pavia, 1501. Died at Rome, 1576. Jerome Cardan was one of the greatest mathematicians of his time. His name is usually applied to the general solution of the cubic equation, although this was really the work of Tartaglia in 1541.

Cardan was astrologer, physician, philosopher and mathematician. HIGHER EQUATIONS SOLVED BY FACTORING 143. Equations of a higher degree than the second can often be solved by factoring and placing each factor equal to zero, as previously explained.

1. Solve, 2x3 – x2 – 15x = 0. . Factoring, x(2x2 – X – 15) = 0.

x(2x + 5) (x – 3) = 0. Place each factor equal to zero, • 2 = 0 2x + 5 = 0

x – 3 = 0 2x = -5

Hence, the three roots are, 0, - and 3.

2. Find the three cube roots of 1.
Let x = the required cube root.
Then 23 = 1
And 23 – 1= 0
Factoring, (x - 1)(x2 + x + 1) = 0
Place each factor equal to zero.
X – 1 = 0

x2 + x + 1 = 0

22 + x = -1
x + x + 1 = -1 + 1 = -1
x + 1 = +1V-3

x = - ] +1V-3 Note that any method of factoring may be used, also that the degree of the equation indicates the number of roots.

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