Answer :
To solve the quadratic equation [tex]\( 3x^2 + x - 5 = 0 \)[/tex], we can use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
For the equation [tex]\( 3x^2 + x - 5 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = -5 \)[/tex]
Step-by-step solution:
1. Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \text{Discriminant} = 1^2 - 4 \cdot 3 \cdot (-5) \][/tex]
[tex]\[ \text{Discriminant} = 1 - (-60) \][/tex]
[tex]\[ \text{Discriminant} = 1 + 60 \][/tex]
[tex]\[ \text{Discriminant} = 61 \][/tex]
2. Find 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 the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and the discriminant:
[tex]\[ x_1 = \frac{{-1 + \sqrt{61}}}{2 \cdot 3} \][/tex]
[tex]\[ x_2 = \frac{{-1 - \sqrt{61}}}{2 \cdot 3} \][/tex]
3. Calculate the exact values of [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]:
[tex]\[ x_1 = \frac{{-1 + \sqrt{61}}}{6} \][/tex]
[tex]\[ x_2 = \frac{{-1 - \sqrt{61}}}{6} \][/tex]
Numerically, these values are approximately:
[tex]\[ x_1 \approx 1.135041612651109 \][/tex]
[tex]\[ x_2 \approx -1.4683749459844424 \][/tex]
4. Round the solutions to 2 decimal places:
[tex]\[ x_1 \approx 1.14 \][/tex]
[tex]\[ x_2 \approx -1.47 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( 3x^2 + x - 5 = 0 \)[/tex] rounded to two decimal places are [tex]\( x_1 = 1.14 \)[/tex] and [tex]\( x_2 = -1.47 \)[/tex].
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
where [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] are the coefficients of the quadratic equation [tex]\( ax^2 + bx + c = 0 \)[/tex].
For the equation [tex]\( 3x^2 + x - 5 = 0 \)[/tex], the coefficients are:
- [tex]\( a = 3 \)[/tex]
- [tex]\( b = 1 \)[/tex]
- [tex]\( c = -5 \)[/tex]
Step-by-step solution:
1. Calculate the discriminant:
[tex]\[ \text{Discriminant} = b^2 - 4ac \][/tex]
Substitute the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]:
[tex]\[ \text{Discriminant} = 1^2 - 4 \cdot 3 \cdot (-5) \][/tex]
[tex]\[ \text{Discriminant} = 1 - (-60) \][/tex]
[tex]\[ \text{Discriminant} = 1 + 60 \][/tex]
[tex]\[ \text{Discriminant} = 61 \][/tex]
2. Find 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 the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and the discriminant:
[tex]\[ x_1 = \frac{{-1 + \sqrt{61}}}{2 \cdot 3} \][/tex]
[tex]\[ x_2 = \frac{{-1 - \sqrt{61}}}{2 \cdot 3} \][/tex]
3. Calculate the exact values of [tex]\( x_1 \)[/tex] and [tex]\( x_2 \)[/tex]:
[tex]\[ x_1 = \frac{{-1 + \sqrt{61}}}{6} \][/tex]
[tex]\[ x_2 = \frac{{-1 - \sqrt{61}}}{6} \][/tex]
Numerically, these values are approximately:
[tex]\[ x_1 \approx 1.135041612651109 \][/tex]
[tex]\[ x_2 \approx -1.4683749459844424 \][/tex]
4. Round the solutions to 2 decimal places:
[tex]\[ x_1 \approx 1.14 \][/tex]
[tex]\[ x_2 \approx -1.47 \][/tex]
Therefore, the solutions to the quadratic equation [tex]\( 3x^2 + x - 5 = 0 \)[/tex] rounded to two decimal places are [tex]\( x_1 = 1.14 \)[/tex] and [tex]\( x_2 = -1.47 \)[/tex].
