Question
\( 1 \leftarrow \) Solve the equation using any method. \[ 4 x^{2}+7=29 x \] Save The sotation set is \( \} \). P. \( \} .93 \). Points: 0 of 2 (Simplify your answer. Type an exact answer, using radicals and \( i \) as needed. Use a comma to separate answers as needed.)
Ask by Ortega Long. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( x = 7 \) and \( x = \frac{1}{4} \).
Solution
To solve the equation \( 4x^{2} + 7 = 29x \), follow these steps:
1. **Bring all terms to one side to form a quadratic equation:**
\[
4x^{2} - 29x + 7 = 0
\]
2. **Use the quadratic formula to find the roots:**
\[
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
\]
Here, \( a = 4 \), \( b = -29 \), and \( c = 7 \).
3. **Calculate the discriminant:**
\[
\Delta = b^{2} - 4ac = (-29)^{2} - 4 \cdot 4 \cdot 7 = 841 - 112 = 729
\]
4. **Find the square root of the discriminant:**
\[
\sqrt{729} = 27
\]
5. **Calculate the two possible solutions:**
\[
x = \frac{29 \pm 27}{8}
\]
- **First solution:**
\[
x = \frac{29 + 27}{8} = \frac{56}{8} = 7
\]
- **Second solution:**
\[
x = \frac{29 - 27}{8} = \frac{2}{8} = \frac{1}{4}
\]
**Final Solutions:**
\[
x = 7, \frac{1}{4}
\]
Reviewed and approved by the UpStudy tutoring team
Like
error msg
Explain 
Simplify this solution The Deep Dive
To solve the equation \( 4x^{2}+7=29x \), we start by rearranging it into a standard quadratic form: \[ 4x^2 - 29x + 7 = 0. \] Next, we can apply the quadratic formula \( x = \frac{-b \pm \sqrt{b^2-4ac}}{2a} \), where \( a = 4 \), \( b = -29 \), and \( c = 7 \). Calculating the discriminant: \[ b^2 - 4ac = (-29)^2 - 4 \cdot 4 \cdot 7 = 841 - 112 = 729. \] Now substituting back into the quadratic formula: \[ x = \frac{29 \pm \sqrt{729}}{2 \cdot 4} = \frac{29 \pm 27}{8}. \] This gives us two solutions: 1. \( x = \frac{56}{8} = 7 \) 2. \( x = \frac{2}{8} = \frac{1}{4} \) Therefore, the solution set is \( \{ 7, \frac{1}{4} \} \).
