To solve the quadratic equation [tex]\( 5x^2 - 7x + 2\sqrt{5} = 0 \)[/tex], we can use the quadratic formula, which is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In this equation, the coefficients are:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = -7\)[/tex]
- [tex]\(c = 2\sqrt{5}\)[/tex]
We will use these coefficients to find the solutions of the quadratic equation.
### Step 1: Calculate the Discriminant
The discriminant ([tex]\(\Delta\)[/tex]) of a quadratic equation [tex]\(ax^2 + bx + c = 0\)[/tex] is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the coefficients [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex] into the discriminant formula:
[tex]\[ \Delta = (-7)^2 - 4(5)(2\sqrt{5}) \][/tex]
[tex]\[ \Delta = 49 - 20 \cdot 2\sqrt{5} \][/tex]
[tex]\[ \Delta = 49 - 40\sqrt{5} \][/tex]
### Step 2: Calculate the Square Root of the Discriminant
Next, we need to find the square root of the discriminant, [tex]\(\Delta\)[/tex]:
[tex]\[ \sqrt{\Delta} = \sqrt{49 - 40\sqrt{5}} \][/tex]
### Step 3: Apply the Quadratic Formula
Now, we can substitute [tex]\(\Delta\)[/tex] and the coefficients [tex]\(a\)[/tex] and [tex]\(b\)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-(-7) \pm \sqrt{49 - 40\sqrt{5}}}{2 \cdot 5} \][/tex]
[tex]\[ x = \frac{7 \pm \sqrt{49 - 40\sqrt{5}}}{10} \][/tex]
### Step 4: Write the Solutions
Finally, we write the two solutions obtained from the quadratic formula:
[tex]\[ x_1 = \frac{7 - \sqrt{49 - 40\sqrt{5}}}{10} \][/tex]
[tex]\[ x_2 = \frac{7 + \sqrt{49 - 40\sqrt{5}}}{10} \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( 5x^2 - 7x + 2\sqrt{5} = 0 \)[/tex] are:
[tex]\[ x = \frac{7 - \sqrt{49 - 40\sqrt{5}}}{10} \][/tex]
[tex]\[ x = \frac{7 + \sqrt{49 - 40\sqrt{5}}}{10} \][/tex]
