Solve by using quadratic formula: $$ 2 x^{2}-5 x-3=0 $$

Question

UpStudy AI · Accepted Answer

To solve the quadratic equation using the quadratic formula, follow these steps:

$$
2x^{2} - 5x - 3 = 0
$$

### Quadratic Formula
The quadratic formula provides the solutions to any quadratic equation of the form $$ ax^{2} + bx + c = 0 $$:

$$
x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}
$$

### Identify the Coefficients
From the given equation, identify the coefficients $$ a $$, $$ b $$, and $$ c $$:

$$
a = 2 \quad
b = -5 \quad
c = -3
$$

### Compute the Discriminant
The discriminant ($$ D $$) determines the nature of the roots and is given by:

$$
D = b^{2} - 4ac
$$

Plugging in the values:

$$
D = (-5)^{2} - 4(2)(-3) = 25 + 24 = 49
$$

### Calculate the Square Root of the Discriminant
$$
\sqrt{D} = \sqrt{49} = 7
$$

### Apply the Quadratic Formula
Substitute $$ b $$, $$ \sqrt{D} $$, and $$ a $$ into the quadratic formula:

$$
x = \frac{-(-5) \pm 7}{2 \times 2} = \frac{5 \pm 7}{4}
$$

### Find the Two Solutions

1. **First Solution ($$ + $$)**:
   $$
   x = \frac{5 + 7}{4} = \frac{12}{4} = 3
   $$

2. **Second Solution ($$ - $$)**:
   $$
   x = \frac{5 - 7}{4} = \frac{-2}{4} = -\frac{1}{2}
   $$

### Conclusion
The solutions to the equation $$ 2x^{2} - 5x - 3 = 0 $$ are:

$$
x = 3 \quad \text{and} \quad x = -\frac{1}{2}
$$
The solutions are $$ x = 3 $$ and $$ x = -\frac{1}{2} $$.

Solve by using quadratic formula: \[ 2 x^{2}-5 x-3=0 \]