Three times the square of a number minus twice the same number is 8. Find all such numbers.
Direct translation:
3x^2 - 2x = 8
3x^2 - 2x - 8 = 0
(x - 2)(3x + 4) = 0
x = 2 or x = -4/3
Thank
eight times the square of a number z minus three
Well, let's call the number "x". According to the jumble of words and numbers you just threw at me, we have the equation 3x^2 - 2x = 8. To solve this, we need to put on our math hats and get to work.
First, let's move that 8 to the other side of the equation, so we have 3x^2 - 2x - 8 = 0. Now, we have ourselves a good old quadratic equation.
To solve a quadratic equation, we can use the quadratic formula. Remember that? It goes like this: x = (-b ± √(b^2 - 4ac)) / (2a). In our case, a = 3, b = -2, and c = -8.
Plugging in those values, we get x = (-(-2) ± √((-2)^2 - 4(3)(-8))) / (2(3)). Simplifying that mess, we get x = (2 ± √(4 + 96)) / 6.
Continuing to simplify, we have x = (2 ± √100) / 6, which further simplifies to x = (2 ± 10) / 6.
Finally, we have two possible solutions. Drumroll, please... x = 12/6 or x = -8/6. Simplifying further, we have x = 2 or x = -4/3.
So, the two numbers which will make this equation hold true are 2 and -4/3.
To find the numbers that satisfy the given equation, let's break down the problem step by step.
Let's assume the number we are looking for is "x". We are given that three times the square of the number minus twice the same number is equal to 8. We can express this in an equation:
3x^2 - 2x = 8
To solve this equation, we need to bring all terms to one side to form a quadratic equation. So, subtract 8 from both sides of the equation:
3x^2 - 2x - 8 = 0
Now, we have a quadratic equation in the standard form that we can solve. To factor or solve the equation, we can use different methods such as factoring, completing the square, or using the quadratic formula.
In this case, factoring may not be straightforward, so let's solve it using the quadratic formula:
The quadratic formula states that for an equation in the form of ax^2 + bx + c = 0, the solution is given by:
x = (-b ± sqrt(b^2 - 4ac)) / (2a)
For our equation, a = 3, b = -2, and c = -8.
Now, substitute these values into the quadratic formula:
x = (-(-2) ± sqrt((-2)^2 - 4 * 3 * (-8))) / (2 * 3)
= (2 ± sqrt(4 + 96)) / 6
= (2 ± sqrt(100)) / 6
= (2 ± 10) / 6
This gives us two possible solutions:
1. When x = (2 + 10) / 6 = 12 / 6 = 2
2. When x = (2 - 10) / 6 = -8 / 6 = -4/3
Therefore, the two numbers that satisfy the given equation are 2 and -4/3.