To solve the quadratic equation [tex]\( 2x^2 - 7x - 5 = 0 \)[/tex], we will use the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
Here, the coefficients are:
- [tex]\( a = 2 \)[/tex]
- [tex]\( b = -7 \)[/tex]
- [tex]\( c = -5 \)[/tex]
First, we need to calculate the discriminant [tex]\( \Delta \)[/tex]:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
Plugging the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex] into the discriminant formula gives:
[tex]\[ \Delta = (-7)^2 - 4 \cdot 2 \cdot (-5) \][/tex]
[tex]\[ \Delta = 49 + 40 \][/tex]
[tex]\[ \Delta = 89 \][/tex]
Since the discriminant ([tex]\( \Delta = 89 \)[/tex]) is positive, we know there are two distinct real solutions.
Next, we find the two solutions using the quadratic formula:
[tex]\[ x_{1,2} = \frac{-b \pm \sqrt{\Delta}}{2a} \][/tex]
Substituting the values of [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( \Delta \)[/tex]:
[tex]\[ x_{1,2} = \frac{7 \pm \sqrt{89}}{4} \][/tex]
Simplifying further, we have two solutions:
[tex]\[ x_1 = \frac{7 + \sqrt{89}}{4} \][/tex]
[tex]\[ x_2 = \frac{7 - \sqrt{89}}{4} \][/tex]
To find the decimal values, compute:
[tex]\[ x_1 ≈ 4.11 \][/tex]
[tex]\[ x_2 ≈ -0.61 \][/tex]
Thus, the values of [tex]\( x \)[/tex] that satisfy the equation [tex]\( 2x^2 - 7x - 5 = 0 \)[/tex] are approximately:
[tex]\[ x_1 = 4.11 \][/tex]
[tex]\[ x_2 = -0.61 \][/tex]
Hence, the solutions to the equation [tex]\( 2x^2 - 7x - 5 = 0 \)[/tex] are [tex]\( x \approx 4.11 \)[/tex] and [tex]\( x \approx -0.61 \)[/tex], both rounded to two decimal places.
