Example 1: Find the square root of 129-72√3. We have from (1) and (2),

1. Rationalize the denominators of

25V3-4V2 10V6-217 V1+x2-VI7V3-5V2' 3√6+27 V1+x2+V1−x2

2. Find the square roots of the following surds: 172-963, 454+126V5, 92-8V60.

3. Multiply 2V-3+3√-2 by 4V-3-5V-2.

7. Find the square roots of the following imaginaries:

5-12√ −1, 47-8√-3, -8√ −1.

8. Find in their simplest forms the roots of

x2-196x2+4=0 and x2+150x2+4761=0.

Ans.

(5√2±4√3), ±(V−3±6√−2).

11. Quadratic Equations.-Any quadratic equation in one unknown quantity, x, can be written

ax2+ bx+c=0,

where a, b, and c are known numbers. Every such equation is satisfied by either of the two values

It is possible to determine some facts about the roots of any quadratic equation without actual solution.

If b2-4ac is less than zero (written b2-4ac<0), the roots are two conjugate imaginaries.

If b2 - 4ac is greater than zero (written b2-4ac>0), the roots are real.

If b2-4ac=0, the roots are equal.

If b2-4ac0 and a perfect square, the roots are rational. The last relation is of no great importance, but the others are very important.

The quantity b2-4ac, since it discriminates among the possible sorts of numbers, is called the discriminant of the equation. The sum and product of the roots of any quadratic equation can be found by immediate inspection of the equation, for

Thus, in the equation 5x2-79.2x+3.28=0,

x1+x2=15.84, x1x,=.656;

the roots are evidently real.

Vb2-4ac
2a

12. Maxima and Minima.-The greatest value an expression can have is called its maximum value; the least, its minimum value. Some problems concerning maxima and minima can be solved by applying the principles of the preceding article.

Example 1: Given y=3+5x-2x2, what is the maximum value y can have if x is real, and what is the value of x that gives this maximum value to y?

Find x in terms of y:

x=1(5±√49− 8y).

Clearly the largest value possible for y, if x is to be real, is 49; and when y has this value, x=. If the value of x had not been required, it would have been sufficient to equate to zero the discriminant of

getting

2x2-5x-3+y=0,

whence

25-8(-3+y)=0,

y=49.

The following examples are similar.

Example 2: What limitation is there on the value of 3x2-10x+16 if x is real?

Putting 3x2-10x+16=y, the discriminant is 12y-92, which is positive only if y is greater than 28. 23 is the minimum value; y may be 23 or anything greater.

Example 3: If 9y-9y2x2+2, and both x and y are real, what limitations are there on the values of x and y?

Solving for 2,

Solving for y,

x=±√ (3y−1) (2 −3y).

y= ±√(x) (1 + x) + 1.

From the first result, it is evident that if y is < or >, x is imaginary; from the second result, if x is <-1 or >1, y is imaginary. y may have any value from to , x any value from - to 1.

Examples.

In each of the following equations decide without solution, and if possible without getting the exact value of the discriminant, what the nature of the roots is, and also give the sum and the product of the roots:

1. 3x2-19x+4=0.

Ans. Roots are real; sum, 19; product, .

2. 5x2+28x-65=0.

Ans. Roots real; sum, -28; product, -13.

3. px2+qx=r, p and r being positive.

4. px2+qx+s=0, p and s being positive.

5. 77x2-139x+62=0.

Find the limitations on the values of the following expressions if x must be real:

6. 7x-3-2x2 and 9x-14—x2.

Ans. First part, 3 maximum value.

7. x2-2x-15 and 2x2-x· 6.

Ans. First part, -16 minimum value.

Ans. First part, - minimum value.

9. For what value of x is (x-a) (b-x) a maximum?

Ans, a+b 2 Find the limitations imposed on x and y by the following relations, if x and y are each real:

10. x2+y2=7y-12. Ans. y<4 and >3, x< and >−1. 11. x2+y2=x+6. Ans. x>−2 and <3, y< and >- §. Ans. y<1 or >5, x unlimited. Ans. x-2 or >4, y unlimited.

12. x2-y2+6y=5.

13. y2+2x+8=x2.

14. Find the limitations on x and the maximum value of the radical in y=(2x+1)±√(x+3)(2−x), if x and y are each real.