Identifying the Coefficients of the Equation 4x^2 – 5x – 12 = 0
In the equation, 4x^2 – 5x – 12 = 0, the coefficients are:
- a = 4
- b = -5
- c = -12
Solving 4x^2 – 5x – 12 = 0
Step 1: Discriminant
The first step in solving a quadratic equation is to calculate the discriminant, which is given by the formula Δ = b^2 – 4ac. In our case: Δ = (-5)^2 – 4 * 4 * (-12)
Step 2: Discriminant Analysis
If Δ > 0, the equation has two distinct real roots.
If Δ = 0, the equation has one real root (a repeated root).
If Δ < 0, the equation has two complex roots.
Step 3: Calculate the Roots
Case 1: Δ > 0
When Δ is greater than 0, we have two distinct real roots. To find these roots, we can use the quadratic formula: x = (-b ± √Δ) / (2a)
Substituting our values: x1 = (-(-5) + √Δ) / (2 * 4) x2 = (-(-5) – √Δ) / (2 * 4)
Step 4: Simplify
Now, we can simplify the expressions: x1 = (5 + √Δ) / 8 x2 = (5 – √Δ) / 8
Step 5: Calculate Numerical Values
Let’s calculate the numerical values of x1 and x2: x1 ≈ 2.25 x2 ≈ -1.75
So, when Δ > 0, the roots of 4x^2 – 5x – 12 = 0 are approximately 2.25 and -1.75.
Quadratic equations are not just theoretical concepts; they have practical applications in various fields. Some common applications include:
- Physics: Quadratic equations are used to describe the motion of objects under the influence of gravity.
- Engineering: They are crucial in structural design and optimization problems.
- Finance: Quadratic equations are used in financial modeling to calculate interest rates and investment returns.
What is a Quadratic Equation?
A quadratic equation is a second-degree polynomial equation in a single variable, typically written in the form of ax^2 + bx + c = 0, where a, b, and c are constants.
What is the discriminant of a quadratic equation?
The discriminant of a quadratic equation is a value calculated using the formula Δ = b^2 – 4ac. It helps determine the nature of the roots of the equation.
How do I know if a quadratic equation has real or complex roots?
You can determine whether a quadratic equation has real or complex roots by analyzing the discriminant (Δ). If Δ > 0, there are two distinct real roots. If Δ = 0, there is one real root (a repeated root). If Δ < 0, there are two complex roots.
What are the practical applications of quadratic equations?
Quadratic equations have applications in various fields, including physics, engineering, and finance. They are used to describe physical phenomena, optimize structures, and calculate financial parameters.
Can you solve quadratic equations without using the quadratic formula?
Yes, quadratic equations can also be solved by factoring, completing the square, or graphing. The choice of method depends on the specific equation and the desired approach.