Questions: 2x^2-5x-3

Transcript text: 5. $2 x^{2}-5 x-3$

Solution

Solution Steps

To solve the quadratic equation $2x^2 - 5x - 3 = 0$, we can use the quadratic formula $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$, where $a$, $b$, and $c$ are the coefficients of the equation $ax^2 + bx + c = 0$.

Solution Approach

Identify the coefficients $a$, $b$, and $c$ from the quadratic equation.
Use the quadratic formula to find the roots of the equation.

Step 1: Identify the Coefficients

The given quadratic equation is $2x^2 - 5x - 3 = 0$. Here, the coefficients are:

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

Step 2: Calculate the Discriminant

The discriminant $D$ is calculated using the formula: \[ D = b^2 - 4ac \] Substituting the values: \[ D = (-5)^2 - 4 \cdot 2 \cdot (-3) = 25 + 24 = 49 \]

Step 3: Find the Roots

Using the quadratic formula: \[ x = \frac{-b \pm \sqrt{D}}{2a} \] Substituting the values: \[ x = \frac{-(-5) \pm \sqrt{49}}{2 \cdot 2} = \frac{5 \pm 7}{4} \] Calculating the two possible values for $x$:

For the positive root: \[ x_1 = \frac{5 + 7}{4} = \frac{12}{4} = 3.0 \]
For the negative root: \[ x_2 = \frac{5 - 7}{4} = \frac{-2}{4} = -0.5 \]

Final Answer

The roots of the equation $2x^2 - 5x - 3 = 0$ are: \[ \boxed{x_1 = 3.0} \] \[ \boxed{x_2 = -0.5} \]

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