There are 5 quadratics below. Four of them have two distinct roots each. The other has only one distinct root; find the value of that root.
4x^2+16x-9
2x^2+80x+400
x^2-6x-9
4x^2-12x+9
-x^2+14x+49
To find the quadratic equation that has only one distinct root, we need to determine which quadratic has a discriminant of zero. The discriminant is the part of the quadratic formula under the square root sign, given by b^2 - 4ac.
Let's calculate the discriminant for each of the given quadratics:
1. For the quadratic 4x^2 + 16x - 9:
a = 4, b = 16, c = -9
Discriminant: (16)^2 - 4(4)(-9)
= 256 + 144
= 400
2. For the quadratic 2x^2 + 80x + 400:
a = 2, b = 80, c = 400
Discriminant: (80)^2 - 4(2)(400)
= 6400 - 3200
= 3200
3. For the quadratic x^2 - 6x - 9:
a = 1, b = -6, c = -9
Discriminant: (-6)^2 - 4(1)(-9)
= 36 + 36
= 72
4. For the quadratic 4x^2 - 12x + 9:
a = 4, b = -12, c = 9
Discriminant: (-12)^2 - 4(4)(9)
= 144 - 144
= 0
5. For the quadratic -x^2 + 14x + 49:
a = -1, b = 14, c = 49
Discriminant: (14)^2 - 4(-1)(49)
= 196 + 196
= 392
Comparing the discriminants, we see that the quadratic equation with a discriminant of zero is:
4x^2 - 12x + 9
Therefore, this quadratic has only one distinct root. To find the value of that root, we can use the formula for the root of a quadratic equation:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our quadratic equation, plugging in the values:
x = (-(-12) ± √((-12)^2 - 4(4)(9))) / (2(4))
= (12 ± √(144 - 144)) / 8
= (12 ± √0) / 8
= (12 ± 0) / 8
= 12 / 8
= 3/2
Hence, the value of the distinct root of the quadratic 4x^2 - 12x + 9 is x = 3/2.