CHAPTER XX

PROPERTIES OF QUADRATIC EQUATIONS

CHARACTER OF THE ROOTS

349. The quadratic equation ax2+bx+c=0 has two roots,

1. If b2-4 ac is positive or equal to zero, the roots are real. If b2-4 ac is negative, the roots are imaginary.

2. If b2-4 ac is a perfect square, the roots are rational. If b2. 4 ac is not a perfect square, the roots are irrational.

3. If b2. 4 ac is zero, the roots are equal.

If b2-4 ac is not zero, the roots are unequal.

350. The expression 62-4 ac is called the discriminant of the equation ax2+ bx+c=0.

Ex. 1. Determine the character of the roots of the equation 3x2-2x-5=0.

The discriminant (-2)2-4.3. (-5)= 64.

=

Hence the roots are real, rational, and unequal.

Ex. 2. Determine the character of the roots of the equation 4x-12x+9=0.

Since (12)24.4.90, the roots are real, rational, and equal. Ex. 3. Prove that the roots of the equation 2+2 px+p1 -q2-2 gr - 20 are rational.

351. The preceding propositions make it possible to deter mine the coefficients so that the roots shall satisfy a given condition.

Ex. 1. Determine the value of m for which the roots of the equation2+x+3=m are equal.

NOTE. This result can be obtained by inspection of the graph of this function, which was discussed in §302.

Ex. 2. Determine the value of m for which the equation (m +5) x2 + 3 mx − 4 (m −5) = 0 has equal roots.

Check. The equations 9 x2 + 12 x + 4 = 0, and x2-12x+36= 0, have equal roots.

EXERCISE 131

Determine, without solution, the character of the roots of each equation:

Determine the value of m for which the roots of the follow

352. If the roots of the equation ax2 + bx + c = 0 are denoted

If the given equation is written in the form a2 + -x+ these results may be expressed as follows:

353. If the coefficient of x2 in a quadratic equation is unity,

(a) The sum of the roots is equal to the coefficient of x with the sign changed.

(b) The product of the roots is equal to the absolute term.

E.g. the sum of the roots of 4 x2 + 5 x − 30 is, their product

354. Formation of equations. If r1 and r2 denote the roots of

b

the quadratic equation ++=0, the equation may be written:

Or factoring,

a

a

To form an equation whose roots are given we may use either (1) or (2).

Ex. 1. Form the equation whose roots are 2 and −3

Ex. 2. Form the equation whose roots are - and — §.

Ex. 3. Form the equation whose roots are 2+√2 and 2-√2.

In each of the following equations determine by inspection the sum and the product of the roots:

1. x2-7x+6=0.

2. x2+8x-2=0.

3. 3x2+5x+3=0.

4. 5x+5 +1=0.

5. x2-(a+b)x+ab = 0.

6. 7x2-x+1=0.

Solve the following equations, and check the answers by forming the sum and the product of the roots:

19. x2-4x+1=0.

20. x2-6x+6=0.

21. x2-6x+4=0.

22. x2+x+1=0.

23. Without solving find the sum of the squares of the roots of the equation ax2 + bx + c = 0.

24. Without solving find the difference of the roots of the equation ax2+ bx + c = 0.

FACTORING OF QUADRATIC EXPRESSIONS

355. Let r, and r2 denote the roots of the equations

=α (x2-[r1+r2]x+r12). (§ 354.)

Or factoring, ax2 + bx + c = a(x − r1)(x − r2).

356. Hence any quadratic expression can be factored. The factors, however, are rational only if the roots of the equation obtained by making the expression equal to zero are rational.