Sure, let's solve the given equation step-by-step.
The original equation given is:
[tex]\[ x = \frac{3x^2 - x}{5} \][/tex]
which needs to be unified with:
[tex]\[ 2x + 15 \][/tex]
### Step 1: Equate the two expressions
Set the two sides equal to each other:
[tex]\[ \frac{3x^2 - x}{5} = 2x + 15 \][/tex]
### Step 2: Clear the fraction
To eliminate the fraction, multiply every term by 5:
[tex]\[ 3x^2 - x = 5(2x + 15) \][/tex]
### Step 3: Distribute on the right side
Expand the expression on the right side:
[tex]\[ 3x^2 - x = 10x + 75 \][/tex]
### Step 4: Move all terms to one side
This will help us form a standard quadratic equation. Subtract [tex]\(10x + 75\)[/tex] from both sides:
[tex]\[ 3x^2 - x - 10x - 75 = 0 \][/tex]
Combine like terms:
[tex]\[ 3x^2 - 11x - 75 = 0 \][/tex]
### Step 5: Solve the quadratic equation
This quadratic equation can be solved using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
In our case, [tex]\(a = 3\)[/tex], [tex]\(b = -11\)[/tex], and [tex]\(c = -75\)[/tex].
### Step 6: Calculate the discriminant
First, compute the discriminant:
[tex]\[ b^2 - 4ac = (-11)^2 - 4 \cdot 3 \cdot (-75) \][/tex]
[tex]\[ b^2 - 4ac = 121 + 900 \][/tex]
[tex]\[ b^2 - 4ac = 1021 \][/tex]
### Step 7: Compute the square root of the discriminant
Find the square root of 1021:
[tex]\[ \sqrt{1021} \approx 31.9539 \][/tex]
### Step 8: Apply the quadratic formula
Substitute into the quadratic formula:
[tex]\[ x = \frac{-(-11) \pm 31.9539}{2 \cdot 3} \][/tex]
[tex]\[ x = \frac{11 \pm 31.9539}{6} \][/tex]
This gives us two solutions:
[tex]\[ x_1 = \frac{11 + 31.9539}{6} \approx \frac{42.9539}{6} \approx 7.1588 \][/tex]
[tex]\[ x_2 = \frac{11 - 31.9539}{6} \approx \frac{-20.9539}{6} \approx -3.4922 \][/tex]
### Conclusion:
The solutions to the given equation are:
[tex]\[ x_1 \approx 7.1588 \][/tex]
[tex]\[ x_2 \approx -3.4922 \][/tex]
