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 that has only one distinct root, we can check the discriminant of each quadratic equation. The discriminant can be calculated using the formula: discriminant = b^2 - 4ac.
Let's calculate the discriminant for each quadratic equation:
1. For 4x^2 + 16x - 9:
Discriminant = (16)^2 - 4(4)(-9) = 256 + 144 = 400
2. For 2x^2 + 80x + 400:
Discriminant = (80)^2 - 4(2)(400) = 6400 - 3200 = 3200
3. For x^2 - 6x - 9:
Discriminant = (-6)^2 - 4(1)(-9) = 36 + 36 = 72
4. For 4x^2 - 12x + 9:
Discriminant = (-12)^2 - 4(4)(9) = 144 - 144 = 0
5. For -x^2 + 14x + 49:
Discriminant = (14)^2 - 4(-1)(49) = 196 + 196 = 392
From the above calculations, we can see that the fourth quadratic equation, 4x^2 - 12x + 9, has a discriminant of 0. This means it has only one distinct root.
To find the value of that root, we can use the formula for finding the root of a quadratic equation: x = (-b ± √(b^2 - 4ac)) / (2a).
Plugging in the values from the fourth quadratic equation, we get:
x = (-(-12) ± √((-12)^2 - 4(4)(9))) / (2(4))
= (12 ± √(144 - 144)) / 8
= (12 ± √(0)) / 8
= (12 ± 0) / 8
Since the discriminant is zero, the ± sign doesn't matter, and we can ignore it. Therefore, the value of the distinct root for the quadratic equation 4x^2 - 12x + 9 is x = 12/8 = 3/2 or 1.5.
To find the quadratic that has only one distinct root, we need to find the discriminant of each quadratic equation. The discriminant can be calculated using the formula b^2 - 4ac.
Let's calculate the discriminant for each quadratic equation:
1. For the quadratic equation 4x^2 + 16x - 9:
Discriminant = (16^2) - (4 * 4 * -9) = 256 + 144 = 400
2. For the quadratic equation 2x^2 + 80x + 400:
Discriminant = (80^2) - (4 * 2 * 400) = 6400 - 3200 = 3200
3. For the quadratic equation x^2 - 6x - 9:
Discriminant = (-6^2) - (4 * 1 * -9) = 36 + 36 = 72
4. For the quadratic equation 4x^2 - 12x + 9:
Discriminant = (-12^2) - (4 * 4 * 9) = 144 - 144 = 0
5. For the quadratic equation -x^2 + 14x + 49:
Discriminant = (14^2) - (4 * -1 * 49) = 196 + 196 = 392
Looking at the results, we can see that the quadratic equation 4x^2 - 12x + 9 has a discriminant of 0. A discriminant of 0 means there is only one distinct root.
To find the value of that root, we can use the formula for finding the root of a quadratic equation:
x = -b / 2a
In our case, the quadratic equation is 4x^2 - 12x + 9.
Using the formula, we can substitute the values of a, b, and c from the quadratic equation:
x = -(-12) / 2 * 4
x = 12 / 8
x = 3/2
Therefore, the quadratic equation 4x^2 - 12x + 9 has a single distinct root of x = 3/2.
check the discriminants: b^2-4ac
If the discriminant is zero, there is only one root.