Question
Solve for \( y \). \[ -30 y=9 y^{2}+25 \] If there is more than one solution, separate them with commas. If there is no solution, click on "No solution."
Ask by Bartlett Donnelly. in the United States
Mar 03,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
\( y = -\frac{5}{3} \)
Solution
We start with the given equation:
-30y = 9y² + 25
Step 1: Rearrange the equation so that one side equals zero.
9y² + 30y + 25 = 0
Step 2: Use the quadratic formula where A = 9, B = 30, and C = 25.
y = [ -B ± √(B² - 4AC) ] / (2A)
Step 3: Calculate the discriminant.
B² - 4AC = 30² - 4(9)(25) = 900 - 900 = 0
Since the discriminant is 0, there is a single solution.
Step 4: Solve for y.
y = -30 / (2 * 9) = -30/18 = -5/3
Thus, the solution is:
y = -5/3
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Simplify this solution The Deep Dive
To solve the equation \( -30y = 9y^2 + 25 \), first, we can rearrange it to set it to zero: \[ 9y^2 + 30y + 25 = 0 \] Next, we can use the quadratic formula \( y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \), where \( a = 9 \), \( b = 30 \), and \( c = 25 \). Calculating the discriminant: \[ b^2 - 4ac = 30^2 - 4 \cdot 9 \cdot 25 = 900 - 900 = 0 \] Since the discriminant is 0, there is exactly one solution: \[ y = \frac{-30 \pm 0}{2 \cdot 9} = \frac{-30}{18} = -\frac{5}{3} \] Thus, the solution is: \[ y = -\frac{5}{3} \]
