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The **roots** for the given **quadratic equation** are **x1 ≈ 2.46625** and **x2 ≈ -1.21625**. For solving the **quadratic equation** **4x² – 5x – 12 = 0** we apply the **quadratic formula**, this method is a steadfast solution strategy when facing any** quadratic equation** of the form **ax² – bx – c = 0**. Below we explain the detailed answer to solve the given quadratic equation.

## Expert Answer

To solve the **quadratic equation 4x² – 5x – 12 = 0**, we can use the **quadratic formula**, which is:

x = (-b ± √(b² – 4ac)) / (2a)

Here, a, b, and c are coefficients from the equation, where a = 4, b = -5, and c = -12.

We start by calculating the** discriminant (Δ)**:

Δ = b² – 4ac

Δ = (-5)² – 4×4×(-12)

Δ = 25 + 192

Δ = 217

The **discriminant (Δ)** is positive, indicating two distinct real solutions.

Using the discriminant, we apply the **quadratic formula**:

x = (5 ± √(217)) / 8

This gives us two solutions for x:

x₁ = (5 + √217) / 8

x₂ = (5 – √217) / 8

For a more precise calculation, we approximate √217:

√217 ≈ 14.73

Substituting this into the solutions, we get:

x1 ≈ (5 + 14.73) / 8

x1 ≈ 19.73 / 8

x1 ≈ 2.46625

and

x2 ≈ (5 – 14.73) / 8

x2 ≈ -9.73 / 8

x2 ≈ -1.21625

## Numerical Result

So, the approximate solutions for the equation are:

x1 ≈ 2.46625 (or x1 ≈ 2.47 if rounded to two decimal places)

x2 ≈ -1.21625 (or x2 ≈ -1.22 if rounded to two decimal places)

## Example

Solve the following **quadratic equation**: `3x² + 6x - 9 = 0`

## Solution

To solve the** quadratic equation 3x² + 6x - 9 = 0**, use the

**quadratic formula**:

**. Here, the coefficients are**

`x = (-b ± √(b² - 4ac)) / (2a)`

`a = 3`

, `b = 6`

, and `c = -9`

.Calculate the discriminant (Δ):

`Δ = b² - 4ac`

`Δ = 6² - 4*3*(-9)`

`Δ = 36 + 108`

`Δ = 144`

Since the discriminant (Δ) is positive, there are two distinct **real solutions**.

Apply the **quadratic** formula:

`x = (-6 ± √144) / (2*3)`

This results in two solutions for x:

`x₁ = (-6 + 12) / 6`

`x₁ = 1`

and

`x₂ = (-6 - 12) / 6`

`x₂ = -3`

So, the solutions to the equation ** 3x² + 6x - 9 = 0** are

**and**

`x₁ = 1`

**. These are the x-values where the parabola represented by the equation would intersect the**

`x₂ = -3`

**x-axis**.