To solve the quadratic equation [tex]\( 3x^2 + 5x - 3 = 0 \)[/tex] using the quadratic formula, let's follow these steps carefully:
The quadratic formula is given by:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Where [tex]\( a = 3 \)[/tex], [tex]\( b = 5 \)[/tex], and [tex]\( c = -3 \)[/tex].
1. Calculate the discriminant:
[tex]\[
\text{Discriminant} = b^2 - 4ac
\][/tex]
Plugging in the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[
\text{Discriminant} = 5^2 - 4(3)(-3) = 25 + 36 = 61
\][/tex]
2. Calculate the two solutions using the quadratic formula:
[tex]\[
x_1 = \frac{-b + \sqrt{\text{Discriminant}}}{2a}
\][/tex]
[tex]\[
x_2 = \frac{-b - \sqrt{\text{Discriminant}}}{2a}
\][/tex]
Substitute [tex]\( a = 3 \)[/tex], [tex]\( b = 5 \)[/tex], and the discriminant calculated:
First solution [tex]\( x_1 \)[/tex]:
[tex]\[
x_1 = \frac{-5 + \sqrt{61}}{2 \times 3} \approx 0.46837494598444235
\][/tex]
Second solution [tex]\( x_2 \)[/tex]:
[tex]\[
x_2 = \frac{-5 - \sqrt{61}}{2 \times 3} \approx -2.135041612651109
\][/tex]
3. Round the solutions to the nearest hundredth:
[tex]\[
x_1 \approx 0.47
\][/tex]
[tex]\[
x_2 \approx -2.14
\][/tex]
Therefore, the solutions to the quadratic equation [tex]\( 3x^2 + 5x - 3 = 0 \)[/tex], rounded to the nearest hundredth, are:
[tex]\[ x = 0.47, -2.14 \][/tex]
