Alright, let's solve the quadratic equation \( 3x^2 + 2x - 4 = 0 \) step-by-step.
1. Identify the coefficients:
The quadratic equation \( 3x^2 + 2x - 4 = 0 \) has the following coefficients:
[tex]\[ a = 3 \][/tex]
[tex]\[ b = 2 \][/tex]
[tex]\[ c = -4 \][/tex]
2. Write down the quadratic formula:
The quadratic formula to find the roots of the equation \( ax^2 + bx + c = 0 \) is:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
3. Calculate the discriminant:
The discriminant \( \Delta \) is given by:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Substituting in the values of \( a \), \( b \), and \( c \):
[tex]\[ \Delta = 2^2 - 4 \cdot 3 \cdot (-4) \][/tex]
[tex]\[ \Delta = 4 + 48 \][/tex]
[tex]\[ \Delta = 52 \][/tex]
4. Calculate the two potential solutions:
Using the quadratic formula, we have two solutions for \( x \):
[tex]\[ x_1 = \frac{{-b + \sqrt{\Delta}}}{2a} \][/tex]
[tex]\[ x_2 = \frac{{-b - \sqrt{\Delta}}}{2a} \][/tex]
Substituting \( b = 2 \), \( \Delta = 52 \), and \( a = 3 \):
[tex]\[ x_1 = \frac{{-2 + \sqrt{52}}}{2 \cdot 3} \][/tex]
[tex]\[ x_2 = \frac{{-2 - \sqrt{52}}}{2 \cdot 3} \][/tex]
5. Simplify the expressions:
[tex]\[ \sqrt{52} \approx 7.21 \][/tex]
[tex]\[ x_1 = \frac{{-2 + 7.21}}{6} \][/tex]
[tex]\[ x_2 = \frac{{-2 - 7.21}}{6} \][/tex]
[tex]\[ x_1 = \frac{{5.21}}{6} \approx 0.87 \][/tex]
[tex]\[ x_2 = \frac{{-9.21}}{6} \approx -1.54 \][/tex]
6. Round the solutions to 2 decimal places (already done in step 5):
The two solutions to the quadratic equation \( 3x^2 + 2x - 4 = 0 \) are:
[tex]\[ x_1 \approx 0.87 \][/tex]
[tex]\[ x_2 \approx -1.54 \][/tex]
So, the solutions to the quadratic equation \( 3x^2 + 2x - 4 = 0 \) are:
[tex]\[ x_1 = 0.87 \][/tex]
[tex]\[ x_2 = -1.54 \][/tex]
