To solve the quadratic equation [tex]\( 7x^2 - 9x + 5 = 0 \)[/tex] using the quadratic formula, we follow these steps:
### Step 1: Identify the coefficients
The standard form of a quadratic equation is [tex]\( ax^2 + bx + c = 0 \)[/tex].
For the given equation [tex]\( 7x^2 - 9x + 5 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 7 \)[/tex]
- [tex]\( b = -9 \)[/tex]
- [tex]\( c = 5 \)[/tex]
### Step 2: Write down the quadratic formula
The quadratic formula is:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
### Step 3: Calculate the discriminant
The discriminant ([tex]\( \Delta \)[/tex]) is part of the quadratic formula under the square root:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substitute the coefficients into the discriminant:
[tex]\[ \Delta = (-9)^2 - 4(7)(5) \][/tex]
[tex]\[ \Delta = 81 - 140 \][/tex]
[tex]\[ \Delta = -59 \][/tex]
### Step 4: Evaluate the square root of the discriminant
Since the discriminant ([tex]\( \Delta \)[/tex]) is negative, this indicates that the solutions involve complex numbers. [tex]\(\sqrt{-59}\)[/tex] can be written as [tex]\( i\sqrt{59} \)[/tex], where [tex]\( i \)[/tex] is the imaginary unit.
### Step 5: Substitute the values into the quadratic formula
Now, substitute [tex]\( b = -9 \)[/tex], [tex]\( a = 7 \)[/tex], and [tex]\( \sqrt{\Delta} = i\sqrt{59} \)[/tex] into the quadratic formula:
[tex]\[ x = \frac{-(-9) \pm i\sqrt{59}}{2 \cdot 7} \][/tex]
[tex]\[ x = \frac{9 \pm i\sqrt{59}}{14} \][/tex]
Thus, the solutions to the quadratic equation [tex]\( 7x^2 - 9x + 5 = 0 \)[/tex] are:
[tex]\[ x = \frac{9 + i\sqrt{59}}{14} \][/tex]
[tex]\[ \text{or} \][/tex]
[tex]\[ x = \frac{9 - i\sqrt{59}}{14} \][/tex]
The filled-out quadratic formula expression is:
[tex]\[ x = \frac{\boxed{9} \pm \boxed{i\sqrt{59}}}{\boxed{14}} \][/tex]
