Imagine a mathematics puzzle that looks like a quadratic equation. Today’s puzzle is 4x^2 – 5x – 12 = 0, and the numbers are hinting that there are answers inside. Two agents are needed to break this code: one with factoring tools and the other with the quadratic formula.
A bright and sharp detective who works with factors tries to break the solution down into two simpler forms. They might show (4x + 3)(x – 4)—two expressions whose dance of multiplication solves the puzzle, like a magician taking a dove apart. By setting all of the factors to zero, they find the first suspects: x = -3/4 and x = 4.
That’s not the end of our story, though. The quadratic formula is like a general truth-seeker that wields algebra’s mighty sword. When we plug the coefficients of our equation into its complicated equation, it gives us the same results, but they are hidden behind mathematical accuracy. They whisper (5 ± √193) / 8 and show the roots with mathematical skill.
Our quadratic finds its “roots,” or the x-values that make the equation true. These are the two places on the graph where it touches the ground. They show that at x = -3/4 and x = 4, our equation’s unseen curve dips down to meet the x-axis, letting everyone know what it’s all about.
Solutions of the equation 4x^2 – 5x – 12 = 0
#1. Solving the Equation:
This quadratic equation can be solved in a number of ways to find its roots or the values of x that make the equation true. Here are two popular ways to do it:
- Factoring: You can try to split the expression 4x^2 – 5x – 12 into two linear expressions whose sum is the same as the original expression. The factors are (4x + 3)(x – 4). If you set each factor to zero, the answers are x = -3/4 and x = 4.
- Quadratic Formula: Any quadratic equation beginning with ax^2 + bx + c = 0 can be solved using the quadratic formula. We have a = 4, b = -5, and c = -12 this time.
When you use the method, you get:
- x = (-b ± √(b^2 – 4ac)) / 2a
- When you change the numbers, you get the same answers as before: x = (-(-5) ± √((-5)^2 – 4 * 4 * -12)) / 2 * 4
- x = (5 ± √193) / 8
#2. Understanding the Roots:
If you know that x = -3/4 and x = 4, you can figure out what they mean by looking at the equation:
- Real roots: As the equation’s two answers are both real numbers, it means that there are two clear points on the x-axis where the parabola it represents meets it.
- Geometric interpretation: In terms of geometry, the roots are the points where the curve of the quadratic function 4x^2 – 5x – 12 meets the x-axis.
#3. Further Analysis:
More math can be done to learn more about the equation and the solutions:
- Discriminant: The differentiator (b^2–4ac) tells us what kind of answers there are. The discriminant is positive (193) in this case, which shows that there are two separate real roots.
- Vertex: Figuring out the parabola’s vertex helps you see how the answers fit into the curve.
In order to determine the solutions to a quadratic problem, how can I apply the finishing the square method?
To use the method of completing the square to find the roots of a quadratic problem, do the following:
#1. Rewrite the equation in standard form:
As long as a, b, and c are constants, the equation should be written as ax^2 + bx + c = 0.
#2. Move the constant term to the right side:
To get to the quadratic terms on the left, add or take away c from both sides: -c = ax^2 + bx
#3. Divide both sides by a (if a ≠ 1):
The equation can now be written as x^2 + (b/a)x = -c/a.
#4. Complete the square:
- Add the square of half of the x value to both sides of the equation.
- Now, the left side will fit into a perfect square.
#5. Factor the perfect square trinomial:
Change the left side to (x + B)^2, where B is x’s half-coefficient.
#6. Isolate x:
- Remember to look at both the positive and negative roots when you take the square root of both sides.
- Solve for x by subtracting B from both sides.
Example:
Solve x^2 + 6x – 7 = 0 using completing the square:
- Rewritten: x^2 + 6x = 7
- Divide by a: x^2 + 6x = 7 (a = 1 in this case)
- Complete the square: x^2 + 6x + 9 = 16 when you add (6/2)^2 = 9.
- Factor: (x + 3)^2 = 16
- Isolate x: x + 3 = ±4
- Solve: x = -3 ± 4
Analyzing the Roots?
To analyze the roots of a quadratic equation, you need to know a lot about their numbers and how they relate to the equation itself. The equation x^2 + 6x – 7 = 0 has roots x = 1 and x = -7. Here are some things you can look at:
#1. Sign and nature of the roots:
- Both roots are real numbers, which means they point to real places on the x-axis where the line and the curve meet.
- One root is the number 1, and the other is the number 7. This shows that the curve touches the x-axis on both sides of the graph.
#2. Discriminant:
- In this case, b^2 – 4ac = 6^2 – 4 * 1 * (-7) = 52, which is a positive number.
- A positive discriminant backs up our results by saying that there are two separate real roots.
#3. Vertex and Axis of Symmetry:
- The parabola’s point is in the middle of the roots, at (-b/2a, f(-b/2a)). The point where the line meets the plane is (-6/2, f(-3)) = (-3, -2).
- The axis of symmetry is a straight line that goes through the point and has the equation x = -b/2a = -3/2.
#4. Intersections with axes:
- As was already said, the x-intercepts are the roots, which are (1, 0) and (-7, 0).
- You can find the y-intercept by making x = 0. This gives you (0, -7). At y = -7, this point is on the y-axis.
#5. Relationship between roots and coefficients:
- Vieta’s methods show how to find the coefficients of a quadratic equation from its roots. In this case, -b/a = -6/1 = -6 is the sum of the roots, which is the same as (1 + (-7) = -6).
- The roots’ product is c/a = -7/1 = -7, which is the same as the roots’ product (1 * (-7) = -7).
What are the four ways to find the roots of a quadratic equation?
There are more than four ways to find the roots of a quadratic equation, but these are the most usual ones:
#1. Factoring:
To use this method, you break the quadratic expression into two linear factors that add up to the original expression. The roots can be found by setting each value to zero. This method works well for problems whose expressions are simple enough to factor, but not all quadratics are simple enough to factor.
#2. Quadratic Formula:
This is a general formula that can be used to find the roots of any quadratic problem. The equation’s factors (a, b, and c) are used to find the roots. This method may be challenging to understand, but it will always work for solving quadratic equations.
#3. Completing the Square:
To use this method, you need to change the solution and add an expression to both sides to make a perfect square trinomial. Taking the square root of both sides makes the problem easier to understand, which lets you find the roots. You can use this way to picture the parabola better than the equation represents.
#4. Graphically:
The roots can be seen by plotting the quadratic function as a parabola and finding the places where it meets the x-axis. This way is less accurate than the others, but it can help you get a general idea of the parabola’s shape and root values.
The best way to do something relies on the equation, the level of accuracy you want, and your taste. Think about how easy the method is, how well you understand the ideas, and whether you need an exact or close answer.
Conclusion
- Equation: 4x^2 – 5x – 12 = 0
- Roots: (-5 ± √193) / 8 ≅ 2.47 & -1.22 (both real numbers)
Properties:
- Discriminant: 193 (positive, showing two separate real roots)
- Vertex: (-5/8, -25/8) ≅ (-0.63, -3.13)
- Axis of symmetry: x = -5/8 ≅ x = -0.63
Solutions:
- You can use factoring, the quadratic formula, or graphs to find it.
- Quadratic formula provides direct solution: x = (-b ± √(b^2 – 4ac)) / 2a
Further Exploration:
- Graphing can help you see roots and parabolas.
- Use Vieta’s formulas to look at the link between roots and coefficients.
Overall:
There are several ways to solve this quadratic problem because it has two evident real roots. Knowing its properties and sources lets you analyze and make sense of it in more depth.
