Sure, let's solve the quadratic equation step by step.
### Step 1: Write the equation in standard quadratic form
Given the equation:
[tex]\[ 3x^2 - 5x + 1 = 4 \][/tex]
Subtract 4 from both sides to get the equation in standard form:
[tex]\[ 3x^2 - 5x + 1 - 4 = 0 \][/tex]
[tex]\[ 3x^2 - 5x - 3 = 0 \][/tex]
### Step 2: Identify the coefficients
In the standard quadratic form [tex]\( ax^2 + bx + c = 0 \)[/tex], identify the coefficients:
[tex]\[ a = 3 \][/tex]
[tex]\[ b = -5 \][/tex]
[tex]\[ c = -3 \][/tex]
### Step 3: 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 values of [tex]\(a\)[/tex], [tex]\(b\)[/tex], and [tex]\(c\)[/tex]:
[tex]\[ \Delta = (-5)^2 - 4(3)(-3) \][/tex]
[tex]\[ \Delta = 25 + 36 \][/tex]
[tex]\[ \Delta = 61 \][/tex]
### Step 4: Apply the quadratic formula
The quadratic formula to find the solutions for [tex]\( ax^2 + bx + c = 0 \)[/tex] is:
[tex]\[ x = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substitute [tex]\(a = 3\)[/tex], [tex]\(b = -5\)[/tex], and [tex]\(\Delta = 61\)[/tex]:
[tex]\[ x = \frac{-(-5) \pm \sqrt{61}}{2(3)} \][/tex]
[tex]\[ x = \frac{5 \pm \sqrt{61}}{6} \][/tex]
### Step 5: Find the two solutions
Calculate the two potential solutions:
[tex]\[ x_1 = \frac{5 + \sqrt{61}}{6} \][/tex]
[tex]\[ x_2 = \frac{5 - \sqrt{61}}{6} \][/tex]
### Step 6: Round the solutions to the nearest hundredth
Convert the values to decimal form and round to the nearest hundredth:
[tex]\[ x_1 \approx 2.14 \][/tex]
[tex]\[ x_2 \approx -0.47 \][/tex]
### Final Solutions
Arrange the solutions from smallest to largest:
[tex]\[ x_1 \approx -0.47 \][/tex]
[tex]\[ x_2 \approx 2.14 \][/tex]
Thus, the solutions to the quadratic equation [tex]\( 3x^2 - 5x + 1 = 4 \)[/tex] are:
[tex]\[ x = -0.47 \][/tex]
and
[tex]\[ x = 2.14 \][/tex]
