Test: Quadratic Equations- 1

x² + 5x - 3x - 15 = 0
x(x + 5) - 3(x + 5) = 0
(x - 3)(x + 5) = 0
⇒ x = 3 or x = -5.

2x² + 6x - 3x - 9 = 0
2x(x + 3) - 3(x + 3) = 0
(x + 3)(2x - 3) = 0
⇒ x = -3 or x = 3/2.

Explanation:

The discriminant of the quadratic equation is (-12)² - 4(3)(10) i.e., 24. As this is positive but not a perfect square, the roots are irrational and unequal.

Explanation:

Any quadratic equation is of the form
x² - (sum of the roots)x + (product of the roots) = 0 ---- (1)
where x is a real variable. As sum of the roots is 13 and product of the roots is -140, the quadratic equation with roots as 20 and -7 is: x² - 13x - 140 = 0.

Explanation:

Sum of the roots and the product of the roots are -20 and 3 respectively.

Explanation:

Let the roots of the equation 2a and 3a respectively.
2a + 3a = 5a = -(- 5/2) = 5/2 => a = 1/2
Product of the roots: 6a² = b/2 => b = 12a²
a = 1/2, b = 3.

Explanation:

Let the two consecutive positive integers be x and x + 1
x² + (x + 1)² - x(x + 1) = 91
x² + x - 90 = 0
(x + 10)(x - 9) = 0 => x = -10 or 9.
As x is positive x = 9
Hence the two consecutive positive integers are 9 and 10.

Explanation:

Let the roots of the quadratic equation be x and 3x.
Sum of roots = -(-12) = 12
a + 3a = 4a = 12 => a = 3
Product of the roots = 3a² = 3(3)² = 27.

Explanation:

Three consecutive even natural numbers be 2x - 2, 2x and 2x + 2.
(2x - 2)² + (2x)² + (2x + 2)² = 1460
4x² - 8x + 4 + 4x² + 8x + 4 = 1460
12x² = 1452 => x² = 121 => x = ± 11
As the numbers are positive, 2x > 0. Hence x > 0. Hence x = 11.
Required numbers are 20, 22, 24.

Explanation:

a² + b² + ab = a² + b² + 2ab - ab
i.e., (a + b)² - ab
from x² - 9x + 20 = 0, we have
a + b = 9 and ab = 20. Hence the value of required expression (9)² - 20 = 61.