Question
Solve the equation \( 20 x^{2}-7 x-6=0 \) Answer: \( x=\square \)
Ask by Vega Craig. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The solutions are \( x = \frac{3}{4} \) and \( x = -\frac{2}{5} \).
Solution
To solve the quadratic equation \( 20x^{2} - 7x - 6 = 0 \), we can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
\]
Here, \( a = 20 \), \( b = -7 \), and \( c = -6 \).
1. **Calculate the discriminant (\( \Delta \)):**
\[
\Delta = b^{2} - 4ac = (-7)^{2} - 4(20)(-6) = 49 + 480 = 529
\]
2. **Take the square root of the discriminant:**
\[
\sqrt{\Delta} = \sqrt{529} = 23
\]
3. **Apply the quadratic formula:**
\[
x = \frac{-(-7) \pm 23}{2 \times 20} = \frac{7 \pm 23}{40}
\]
This gives two possible solutions:
\[
x = \frac{7 + 23}{40} = \frac{30}{40} = \frac{3}{4}
\]
\[
x = \frac{7 - 23}{40} = \frac{-16}{40} = -\frac{2}{5}
\]
**Answer:**
\[
x = \frac{3}{4},\ -\frac{2}{5}
\]
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Simplify this solution Extra Insights
To solve the equation \( 20 x^{2}-7 x-6=0 \), we can use the quadratic formula, which is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] Here, \( a = 20 \), \( b = -7 \), and \( c = -6 \). Calculating the discriminant: \[ b^2 - 4ac = (-7)^2 - 4 \cdot 20 \cdot (-6) = 49 + 480 = 529 \] Now, substituting back into the quadratic formula: \[ x = \frac{-(-7) \pm \sqrt{529}}{2 \cdot 20} = \frac{7 \pm 23}{40} \] This gives us two possible solutions: 1. \( x = \frac{30}{40} = \frac{3}{4} \) 2. \( x = \frac{-16}{40} = -\frac{2}{5} \) Thus, the solutions are: \( x = \frac{3}{4} \) or \( x = -\frac{2}{5} \) Answer: \( x=\frac{3}{4}, -\frac{2}{5} \)
