**Find the sum and product of the roots of the quadratic equation:**

- x
^{2}– 5x + 8 = 0 - 3a
^{2}– 10a – 5 = 0 - 8m
^{2}– m = 2 - 6k
^{2}– 3 = 0 - pr
^{2}= r – 5 - x
^{2}+ (ab)x + (a + b) = 0

**Quadratic Equation – Exercise 9.7 – Solutions:**

**Find the sum and product of the roots of the quadratic equation:**

**x**^{2}– 5x + 8 = 0

Solution:

Let m and n are the roots of the quadratic equations x^{2} – 5x + 8 = 0 which is of the type ax^{2} + bx + c = 0, then, where a = 1 , b = -5 , c = 8

Formula to find the sum of the roots and product of the roots are m+n = ^{-b}/_{a} and mn = ^{c}/_{a} respectively.

Therefore, m + n = ^{-b}/_{a} = ^{-(-5)}/_{1} = 5 and also, mn = ^{c}/_{a} = ^{8}/_{1} = 8

Thus, the sum of the roots of the quadratic equation x^{2} – 5x + 8 = 0 is 5 and the product of the roots of the quadratic equation x^{2} – 5x + 8 = 0 is 8.

**3a**^{2}– 10a – 5 = 0

Solution:

Let m and n are the roots of the quadratic equations 3a^{2} – 10a – 5 = 0 which is of the type ax^{2} + bx + c = 0, then, where a = 3 , b = -10 , c = -5

Formula to find the sum of the roots and product of the roots are m+n = ^{-b}/_{a} and mn = ^{c}/_{a} respectively.

Therefore, m + n = ^{-b}/_{a} = ^{-(-10)}/_{3} = ^{10}/_{3 }and also, mn = ^{c}/_{a} = ^{-5}/_{3}

Thus, the sum of the roots of the quadratic equation 3a^{2} – 10a – 5 = 0 is ^{10}/_{3} and the product of the roots of the quadratic equation 3a^{2} – 10a – 5 = 0 is ^{-5}/_{3}.

**8m**^{2}– m = 2

Solution:

The given equation 8m^{2} – m = 2 can be written as 8m^{2} – m – 2 = 0

Let m and n are the roots of the quadratic equations 8m^{2} – m – 2 = 0 which is of the type ax^{2} + bx + c = 0, then, where a = 8 , b = -1 , c = -2

Formula to find the sum of the roots and product of the roots are m+n = ^{-b}/_{a} and mn = ^{c}/_{a} respectively.

Therefore, m + n = ^{-b}/_{a} = ^{-(-1)}/_{8} = ^{1}/_{8} and also, mn = ^{c}/_{a} = ^{-2}/_{8} = –^{1}/_{4}

Thus, the sum of the roots of the quadratic equation 8m^{2} – m – 2 = 0 is ^{1}/_{8} and the product of the roots of the quadratic equation 8m^{2} – m – 2 = 0is –^{1}/_{4}.

**6k**^{2}– 3 = 0

Solution:

The given equation 6k^{2} – 3 = 0 can be written as 6k^{2} + 0.k – 3 = 0

Let m and n are the roots of the quadratic equations 6k^{2} – 3 = 0 which is of the type ax^{2} + bx + c = 0, then, where a = 6 , b = 0 , c = -3

^{-b}/_{a} and mn = ^{c}/_{a} respectively.

Therefore, m + n = ^{-b}/_{a} = ^{-(0)}/_{6} = 0 and also, mn = ^{c}/_{a} = ^{-3}/_{6} = –^{1}/_{2}

Thus, the sum of the roots of the quadratic equation 6k^{2} – 3 = 0 is 0 and the product of the roots of the quadratic equation 6k^{2} – 3 = 0 is –^{1}/_{2}.

**pr**^{2}= r – 5

Solution:

The given equation pr^{2} = r – 5 can be written as pr^{2} – r + 5 = 0

Let m and n are the roots of the quadratic equations pr^{2} – r + 5 = 0 which is of the type ax^{2} + bx + c = 0, then, where a = p , b = -1 , c = 5

^{-b}/_{a} and mn = ^{c}/_{a} respectively.

Therefore, m + n = ^{-b}/_{a} = ^{-(-1)}/_{p} = ^{1}/_{p} and also, mn = ^{c}/_{a} = ^{5}/_{p}

Thus, the sum of the roots of the quadratic equation pr^{2} – r + 5 = 0 is ^{1}/_{p} and the product of the roots of the quadratic equation pr^{2} – r + 5 = 0 is ^{5}/_{p}.

**x**^{2}+ (ab)x + (a + b) = 0

Solution:

Let m and n are the roots of the quadratic equations x^{2} + (ab)x + (a + b) = 0 which is of the type ax^{2} + bx + c = 0, then, where a = 1 , b = (ab) , c = (a+b)

^{-b}/_{a} and mn = ^{c}/_{a} respectively.

Therefore, m + n = ^{-b}/_{a} = ^{-ab}/_{1} = -ab and also, mn = ^{c}/_{a} = ^{(a+b)}/_{1} = (a + b)

Thus, the sum of the roots of the quadratic equation x^{2} + (ab)x + (a + b) = 0 is (ab) and the product of the roots of the quadratic equation x^{2} + (ab)x + (a + b) = 0 is (a + b).