Grade 10 Math - Can you help me?

Question

1. For what values of c will the equation x^2 + 14x +c =0 have

a) 2 distinct real roots b) 2 equal real roots c) no real roots

2. For what values of b will the equation x^2 + bx +25 =0 have

a) 2 distinct real roots b) 2 equal real roots c) no real roots

Answer 1

You have to use the discriminant. The formula of the discriminant is b^2 - 4ac. a is the coefficient with x^2, b is the coefficient with x and c is the remaining coefficient.

An equation with 2 distinct real roots has a discriminant greater than 0.

An equation with 2 equal real roots has a discriminant equal to 0.

An equation with no real roots has a discriminant less than 0.

1. Let us first solve for the discriminant.

a = 1

b = 14

c = c

a) b^2 - 4ac > 0

   196 - 4c > 0

    196 > 4c

    c < 49

b) b^2 - 4ac = 0

    196 - 4c = 0

    c = 49

c) b^2 - 4ac < 0

   196 - 4c <0

   196 < 4c

   c > 49

2. a = 1

    b = b

    c = 25

Substituting...

D = b^2- 4ac

D = b^2 - 100

a. D > 0

   b^2 - 100 > 0

   b^2 > 100

   b < -10 or b>10  

b. D = 0

   b^2 - 100 = 0

   b^2 = 100

   b = positive or negative 10

c. D < 0

   b^2 - 100 < 0

   (b+10)(b-10) < 0

   -10 < b < 10

(The value of b is between -10 and 10.)

All of the above ways are correct ways of solving for the values asked. I just solved them differently. You can look for the method which you prefer the most. Either way, just use the discriminant and remember the cases (D > 0 --> 2 real roots and so on) and solve for whatever you have to solve from then on. Good luck.