Solve using the quadratic formula. Round your answers to the nearest thousandth as needed. $$ 5 x^{2}-12 x=4 $$

Question

UpStudy AI · Accepted Answer

1. Write the equation in standard form:
   $$
   5x^2 - 12x - 4 = 0
   $$

2. Identify the coefficients:
   $$
   a = 5,\quad b = -12,\quad c = -4
   $$

3. Apply the quadratic formula:
   $$
   x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
   $$
   Substituting the coefficients:
   $$
   x = \frac{-(-12) \pm \sqrt{(-12)^2 - 4(5)(-4)}}{2(5)}
   = \frac{12 \pm \sqrt{144 + 80}}{10}
   $$

4. Simplify under the square root:
   $$
   \sqrt{144+80} = \sqrt{224}
   $$
   Notice that $$224 = 16 \times 14$$, so:
   $$
   \sqrt{224} = 4\sqrt{14}
   $$

5. Substitute back into the formula:
   $$
   x = \frac{12 \pm 4\sqrt{14}}{10} = \frac{6 \pm 2\sqrt{14}}{5}
   $$

6. Find the approximate numerical values by evaluating $$\sqrt{14} \approx 3.742$$:

- For the positive root:
     $$
     x \approx \frac{6 + 2(3.742)}{5} = \frac{6 + 7.484}{5} = \frac{13.484}{5} \approx 2.697
     $$
   
   - For the negative root:
     $$
     x \approx \frac{6 - 2(3.742)}{5} = \frac{6 - 7.484}{5} = \frac{-1.484}{5} \approx -0.297
     $$

The solutions, rounded to the nearest thousandth, are:
$$
x \approx 2.697 \quad \text{and} \quad x \approx -0.297
$$
The solutions are approximately $$ x \approx 2.697 $$ and $$ x \approx -0.297 $$.

Solve using the quadratic formula. Round your answers to the nearest thousandth as needed. \( 5 x^{2}-12 x=4 \)

Upstudy AI Solution

Answer

Solution

Beyond the Answer

Related Questions

Latest Algebra Questions