Answer :
To solve the quadratic equation [tex]\(2x^2 + x - 4 = 0\)[/tex], we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a = 2\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -4\)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients:
[tex]\[ a = 2, \quad b = 1, \quad c = -4 \][/tex]
2. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values:
[tex]\[ \Delta = 1^2 - 4 \cdot 2 \cdot (-4) \][/tex]
[tex]\[ \Delta = 1 - (-32) \][/tex]
[tex]\[ \Delta = 1 + 32 \][/tex]
[tex]\[ \Delta = 33 \][/tex]
3. Compute the square root of the discriminant:
[tex]\[ \sqrt{\Delta} = \sqrt{33} \][/tex]
4. Find the roots using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute in the values:
[tex]\[ x = \frac{-1 \pm \sqrt{33}}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{33}}{4} \][/tex]
5. Calculate the two solutions:
- For the positive root:
[tex]\[ x_1 = \frac{-1 + \sqrt{33}}{4} \][/tex]
[tex]\[ x_1 \approx \frac{-1 + 5.744}{4} \][/tex]
[tex]\[ x_1 \approx \frac{4.744}{4} \][/tex]
[tex]\[ x_1 \approx 1.186 \][/tex]
- For the negative root:
[tex]\[ x_2 = \frac{-1 - \sqrt{33}}{4} \][/tex]
[tex]\[ x_2 \approx \frac{-1 - 5.744}{4} \][/tex]
[tex]\[ x_2 \approx \frac{-6.744}{4} \][/tex]
[tex]\[ x_2 \approx -1.686 \][/tex]
### Final Answer:
- The solutions to the equation [tex]\(2x^2 + x - 4 = 0\)[/tex] are:
[tex]\[ x_1 \approx 1.186 \quad \text{and} \quad x_2 \approx -1.686 \][/tex]
- The discriminant is [tex]\(33\)[/tex], indicating two distinct real roots.
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
where [tex]\(a = 2\)[/tex], [tex]\(b = 1\)[/tex], and [tex]\(c = -4\)[/tex].
### Step-by-Step Solution:
1. Identify the coefficients:
[tex]\[ a = 2, \quad b = 1, \quad c = -4 \][/tex]
2. Calculate the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging in the values:
[tex]\[ \Delta = 1^2 - 4 \cdot 2 \cdot (-4) \][/tex]
[tex]\[ \Delta = 1 - (-32) \][/tex]
[tex]\[ \Delta = 1 + 32 \][/tex]
[tex]\[ \Delta = 33 \][/tex]
3. Compute the square root of the discriminant:
[tex]\[ \sqrt{\Delta} = \sqrt{33} \][/tex]
4. Find the roots using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute in the values:
[tex]\[ x = \frac{-1 \pm \sqrt{33}}{2 \cdot 2} \][/tex]
[tex]\[ x = \frac{-1 \pm \sqrt{33}}{4} \][/tex]
5. Calculate the two solutions:
- For the positive root:
[tex]\[ x_1 = \frac{-1 + \sqrt{33}}{4} \][/tex]
[tex]\[ x_1 \approx \frac{-1 + 5.744}{4} \][/tex]
[tex]\[ x_1 \approx \frac{4.744}{4} \][/tex]
[tex]\[ x_1 \approx 1.186 \][/tex]
- For the negative root:
[tex]\[ x_2 = \frac{-1 - \sqrt{33}}{4} \][/tex]
[tex]\[ x_2 \approx \frac{-1 - 5.744}{4} \][/tex]
[tex]\[ x_2 \approx \frac{-6.744}{4} \][/tex]
[tex]\[ x_2 \approx -1.686 \][/tex]
### Final Answer:
- The solutions to the equation [tex]\(2x^2 + x - 4 = 0\)[/tex] are:
[tex]\[ x_1 \approx 1.186 \quad \text{and} \quad x_2 \approx -1.686 \][/tex]
- The discriminant is [tex]\(33\)[/tex], indicating two distinct real roots.
