If four times the square of a positive number is added to the product of -29 and the number, the result is 63. Find the number
Let x be the number.
Write everything in math expression using x.
Four time the square of a number: 4x^2
Is added to the product of -29 and the number: + (-29)x
The result is 63: = 63
Combining them, we have
4x^2 + (-29)x = 63
or
4x^2 - 29x - 63 = 0
We have a quadratic equation here. Good thing is, it's factorable:
(4x + 7)(x - 9) = 0
4x + 7 = 0
x = -7/4
x - 9 = 0
x = 9
Since in the problem, the number is 'positive', we take x = 9.
Hope this helps~ `u`
Thank you this makes sense now!
Let's start by setting up the equation. Let's assume the positive number is "x".
According to the problem, four times the square of the positive number is added to the product of -29 and the number, resulting in 63.
This can be written as: 4x^2 + (-29)x = 63.
To solve this equation, we need to bring all the terms to one side to obtain a quadratic equation in standard form.
So, the equation becomes: 4x^2 - 29x - 63 = 0.
Now, we can solve this quadratic equation to find the value(s) of x. We can either factor, complete the square, or use the quadratic formula.
Let's use factoring to solve this equation:
To factor the equation, we need to find two numbers that multiply to give -252 (the product of 4(-63)) and add up to -29.
After some trial and error, we find that the two numbers are -36 and 7.
So, factoring the equation, we have: (4x + 7)(x - 9) = 0.
Setting each factor equal to zero, we get two possible solutions:
1. 4x + 7 = 0, which gives x = -7/4.
2. x - 9 = 0, which gives x = 9.
However, since we are looking for a positive number, the only valid solution is x = 9.
Therefore, the positive number is 9.
To solve this problem, let's break it down into steps:
Step 1: Let's assume the positive number as "x".
Step 2: We can set up the equation based on the given information: 4x^2 + (-29)*x = 63.
Step 3: Simplify the equation by multiplying -29 and x: 4x^2 - 29x = 63.
Step 4: Bring all terms to one side to form a quadratic equation: 4x^2 - 29x - 63 = 0.
Step 5: Solve this quadratic equation by factoring, completing the square, or using the quadratic formula.
Since factoring may not be straightforward in this case, let's use the quadratic formula:
The quadratic formula states that for any equation in the form ax^2 + bx + c = 0, the solutions can be found using the formula:
x = (-b ± √(b^2 - 4ac)) / (2a)
For our equation 4x^2 - 29x - 63 = 0, we have a = 4, b = -29, and c = -63.
Using the quadratic formula, we can calculate the solutions for x:
x = (-(-29) ± √((-29)^2 - 4*4*(-63)))/(2*4)
x = (29 ± √(841 + 1008)) / 8
x = (29 ± √1849) / 8
Taking the positive square root (since we are looking for a positive number), we have:
x = (29 + √1849) / 8 or x = (29 - √1849) / 8
Calculating the square root of 1849, we get:
x = (29 + 43) / 8 or x = (29 - 43) / 8
x = 72 / 8 or x = -14 / 8
x = 9 or x = -1.75
Since we are looking for a positive number, the solution is x = 9.
Therefore, the positive number in this case is 9.