To solve the quadratic equation [tex]\(5x^2 - 8x = 3\)[/tex], we can start by putting it in the standard form of [tex]\(ax^2 + bx + c = 0\)[/tex]:
[tex]\[5x^2 - 8x - 3 = 0\][/tex]
### Step-by-Step Solution:
1. Attempt to Factor the Equation:
Let's first try to factor the quadratic equation:
[tex]\[5x^2 - 8x - 3 = 0\][/tex]
To factor this, we look for two numbers that multiply to [tex]\(a \cdot c = 5 \cdot -3 = -15\)[/tex] and add to [tex]\(b = -8\)[/tex]. These two numbers would be used to split the middle term, but in this case, it's challenging to find such numbers. Therefore, factoring doesn't seem straightforward here.
2. Use the Quadratic Formula:
Since factoring is not easily working out, we use the quadratic formula:
[tex]\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\][/tex]
For our equation [tex]\(5x^2 - 8x - 3 = 0\)[/tex]:
- [tex]\(a = 5\)[/tex]
- [tex]\(b = -8\)[/tex]
- [tex]\(c = -3\)[/tex]
Substitute these values into the formula:
[tex]\[x = \frac{-(-8) \pm \sqrt{(-8)^2 - 4 \cdot 5 \cdot (-3)}}{2 \cdot 5}\][/tex]
Simplify step by step:
1. Calculate the discriminant:
[tex]\[b^2 - 4ac = (-8)^2 - 4 \cdot 5 \cdot (-3) = 64 + 60 = 124\][/tex]
2. Substitute the discriminant back into the quadratic formula:
[tex]\[x = \frac{8 \pm \sqrt{124}}{10}\][/tex]
3. Simplify [tex]\(\sqrt{124}\)[/tex]:
[tex]\[\sqrt{124} = \sqrt{4 \cdot 31} = 2\sqrt{31}\][/tex]
4. Substitute back into the equation:
[tex]\[x = \frac{8 \pm 2\sqrt{31}}{10} = \frac{8}{10} \pm \frac{2\sqrt{31}}{10}\][/tex]
5. Simplify the fractions:
[tex]\[x = \frac{4}{5} \pm \frac{\sqrt{31}}{5}\][/tex]
Thus, the two solutions are:
[tex]\[x = \frac{4}{5} - \frac{\sqrt{31}}{5}, \quad x = \frac{4}{5} + \frac{\sqrt{31}}{5}\][/tex]
### Final Answer:
[tex]\[x = \frac{4}{5} - \frac{\sqrt{31}}{5}, \quad x = \frac{4}{5} + \frac{\sqrt{31}}{5}\][/tex]
These are the solutions to the quadratic equation [tex]\(5x^2 - 8x - 3 = 0\)[/tex].
