Solving quadratic equations is a fundamental skill in algebra, and understanding how to approach them can greatly enhance your mathematical proficiency. In this comprehensive guide, we will delve into solving the quadratic equation 4x² – 5x – 12 = 0 using various methods, including the quadratic formula, factoring, and completing the square. By the end of this article, you’ll have a clear understanding of these techniques and be able to apply them confidently to similar problems.
Understanding the Equation 4x² – 5x – 12 = 0
The given equation is a standard quadratic equation of the form ax² + bx + c = 0, where:
- a = 4
- b = -5
- c = -12
This equation represents a parabola that opens upwards (since a > 0) and intersects the x-axis at the solutions to the equation. Our goal is to find the values of x that satisfy this equation.
Method 1: Using the Quadratic Formula
The quadratic formula is a universal method for solving any quadratic equation. It is expressed as:
x = (-b ± √(b² – 4ac)) / 2a
Step-by-Step Solution
- Identify the coefficients: From the equation 4x² – 5x – 12 = 0, we have:
- a = 4
- b = -5
- c = -12
- Calculate the discriminant: The discriminant (Δ) is the part under the square root in the quadratic formula and is calculated as:Δ = b² – 4ac
Substituting the values:
Δ = (-5)² – 4(4)(-12) = 25 + 192 = 217
- Apply the quadratic formula: Substitute the values of a, b, and Δ into the quadratic formula:x = [5 ± √217] / 8
- Simplify the expression: Approximating the square root of 217:√217 ≈ 14.73
Therefore, the two solutions are:
-
- x₁ = (5 + 14.73) / 8 ≈ 19.73 / 8 ≈ 2.47
- x₂ = (5 – 14.73) / 8 ≈ -9.73 / 8 ≈ -1.22
Thus, the solutions to the equation are approximately x ≈ 2.47 and x ≈ -1.22.
Method 2: Factoring the Quadratic Equation
Factoring involves expressing the quadratic equation as a product of two binomials. However, not all quadratic equations are factorable using integers. In this case, let’s attempt to factor 4x² – 5x – 12 = 0.
Step-by-Step Solution
- Multiply a and c: Multiply the coefficient of x² (a) by the constant term (c):
4 * (-12) = -48
- Find two numbers that multiply to -48 and add to -5: The numbers -8 and 6 satisfy these conditions.
- Rewrite the middle term: Express -5x as the sum of -8x and 6x:4x² – 8x + 6x – 12 = 0
- Factor by grouping: Group the terms and factor each group:(4x² – 8x) + (6x – 12) = 0
4x(x – 2) + 6(x – 2) = 0
- Factor out the common binomial: Factor out (x – 2):
(x – 2)(4x + 6) = 0
- Solve for x: Set each factor equal to zero:
- x – 2 = 0 → x = 2
- 4x + 6 = 0 → x = -6/4 = -3/2 Thus, the solutions are x = 2 and x = -3/2.
Method 3: Completing the Square
Completing the square is a method that involves rewriting the quadratic equation in the form (x – p)² = q, which can then be solved easily.
Step-by-Step Solution
- Divide the equation by a: If a ≠ 1, divide the entire equation by a to simplify:
x² – (5/4)x – 3 = 0
- Move the constant term: Move the constant term to the other side of the equation:
x² – (5/4)x = 3
- Add the square of half the coefficient of x: Take half of -5/4, square it, and add it to both sides:
(5/8)² = 25/64