Learn how to solve the quadratic equation 4x^2 + 5x – 12 = 0 / 4x 2 5x 12 0 Solving step by step. Discover the solutions and the quadratic formula in this comprehensive guide to quadratic equations.
Quadratic equations are a fundamental concept in algebra and mathematics, with various real-world applications. In this blog post, we will explore how to solve a specific quadratic equation: 4x^2 + 5x – 12 = 0. We’ll take you through the step-by-step process to find the solutions and understand the principles behind it.
Understanding the Quadratic Equation
A quadratic equation is a polynomial equation of the second degree, typically written in the form ax^2 + bx + c = 0. In our case, the equation is 4x^2 + 5x – 12 = 0, where a = 4, b = 5, and c = -12.
Step 1: Identify Coefficients
Before we can solve the quadratic equation, we need to identify the coefficients a, b, and c. In our equation, they are:
- a = 4
- b = 5
- c = -12
Step 2: Use the Quadratic Formula
The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
x = (-b ± √(b^2 – 4ac)) / (2a)
Using this formula, we can find the values of x that satisfy the equation.
Step 3: Substitute the Coefficients
Now, let’s substitute the coefficients from our equation into the quadratic formula:
x = (-5 ± √(5^2 – 4 * 4 * (-12))) / (2 * 4)
Step 4: Calculate the Discriminant
The discriminant, represented by Δ (Delta), is the value inside the square root in the quadratic formula. It determines the nature of the solutions:
- If Δ > 0, there are two distinct real solutions.
- If Δ = 0, there is one real solution (a repeated root).
- If Δ < 0, there are no real solutions (complex roots).
In our equation, Δ = 5^2 – 4 * 4 * (-12) = 25 + 192 = 217.
Step 5: Calculate the Solutions
Now, we can calculate the solutions by substituting Δ and the coefficients back into the quadratic formula:
x₁ = (-5 + √217) / (2 * 4) ≈ 1.25 x₂ = (-5 – √217) / (2 * 4) ≈ -3
Step 6: Interpret the Results
We have found two solutions for our quadratic equation:
- x₁ ≈ 1.25
- x₂ ≈ -3
These are the x-values that satisfy the equation 4x^2 + 5x – 12 = 0.
Conclusion
Solving quadratic equations is an essential skill in mathematics, and it can be applied to various real-world problems. In this example, we used the quadratic formula to find the solutions for the equation 4x^2 + 5x – 12 = 0. By following these steps and understanding the principles behind them, you can confidently tackle quadratic equations of your own and gain a deeper appreciation for algebraic problem-solving.
