Mathematics always poses problems that, at face value, are difficult to solve. Among these are quadratic and algebraic equations which form the basics of algebra which is important when developing strong foundations on mathematics. The quadratic equation 4x ^ 2 – 5x – 12 = 0 is an example of this case: The coefficients a =4, b=-5, and c=-12. When the problem is presented to you, one might think it is difficult to solve but in actual practice, it is not. This article will guide you through the process of solving this quadratic equation step by step, using two reliable methods: factoring and secondly by the use of the quadratic formula.
What Are Quadratic Equations?
However, for one to fully comprehend the solution of the quadratic equation, it is imperative that one knows what a quadratic equation is. A quadratic equation is a type of polynomial equation with a maximum degree of two- that is, has at least one term that contains a variable squared.
The standard form of a quadratic equation is:
ax2+bx+c=0ax2+bx+c=0
As for the format, let me explain that aa, bb, and cc are constants whereas xx is the variable. The equation for which we are trying to find roots is 4x^2 – 5x – 12, which easily fits into this format as a=4a=4, b=−5b=−5, and c=−12c=−12.
The solutions of a quadratic equation are called the roots of the equation, and the task to find these values of xx. Such solutions can be computed using many approaches including factoring and, in this case, the quadratic formula to which we’ll devote major attention.
Characteristics of the 4x ^ 2 – 5x – 12 = 0 Quadratic Formula
Specifically, there are peculiarities that differ from other types of equations and can be helpful in studying the nature of the equation-solving process. For the equation 4x ^ 2 – 5x – 12 = 0, let’s explore some of these characteristics:
- Parabolic Shape: The graph of any quadratic equation is a parabola which is U-shaped. This is the measure of the steepness (slope) of this curve. Depending on the sign of aa, this curve opens up or down. In our case, a is equal to four so it is positive, therefore, the parabola is a rising parabola.
- Vertex: The trend of the curve at the vertex; the vertex of the parabola is the highest or the lowest point of the curve. First, let me draw an identically shaped upward-opening parabola on one of the coordinate planes: As I said, for an upward-opening parabola such as this one, the vertex, represented by the point (h, k), is the minimum point.
- Axis of Symmetry: The parabola is also equidistant from the vertical axis running through the midpoint of the vertex, called, the axis of symmetry. This line can be found using the formula x = – 2ab/x substituting this value of x gives.
- Roots: The next two inputs represent the locations where the parabola crosses the x-axis of the coordinate plane, which gives the solutions of the chosen quadratic equation. These are also referred to as the solutions of the equation or the coefficients or the zeros or roots of the said equation.
Approaches of Solving the Quadratic Equation 4x ^ 2 – 5x – 12 = 0
Having introduced the task, let us move to the concepts of two methods for solving the quadratic equation 4x^2-5x-12=0, namely factoring and the quadratic formula.
1. Factoring Method
As mentioned above, factoring is one of the easiest techniques and most closely related to students’ understanding that can be used to solve a quadratic equation if such an equation is factored. The general concept is to factorize the quadratic equation as the product of two binomials of the first degree, then find the values of x for which each of the two binomials is equal to zero.
Let’s solve 4x ^ 2 – 5x – 12 = 0 by factoring:
1: Multiply aa and cc:
- Here, a=4a=4 and c=−12c=−12, so 4×−12=−484×−12=−48.
2: Find two numbers that multiply to -48 and add to b=−5b=−5:
- The numbers -8 and 6 fit this requirement, as −8×6=−48−8×6=−48 and −8+6=−2−8+6=−2.
3: Rewrite the middle term using these numbers:
- The equation becomes 4×2−8x+6x−12=04×2−8x+6x−12=0.
4: Factor by grouping:
- Group the terms: (4×2−8x)+(6x−12)=0(4×2−8x)+(6x−12)=0.
- Factor out the greatest common factor from each group: 4x(x−2)+6(x−2)=04x(x−2)+6(x−2)=0.
- Factor out the common binomial: (4x+6)(x−2)=0(4x+6)(x−2)=0.
5: Solve for xx:
- Set each factor equal to zero: 4x+6=04x+6=0 or x−2=0x−2=0.
- Solve each equation:
- 4x+6=04x+6=0 leads to x=−32x=−23.
- x−2=0x−2=0 leads to x=2x=2.
Thus, the solutions to the equation 4x^2 – 5x – 12 = 0 are x=−32x=−23 and x=2x=2.
2. Quadratic Formula
The quadratic formula is a universal method that can solve any quadratic equation, even those that are not easily factorable. The formula is:
x=−b±b2−4ac2ax=2a−b±b2−4ac
Let’s use this formula to solve 4x^2 – 5x – 12 = 0.
Step 1: Identify the coefficients aa, bb, and cc:
- Here, a=4a=4, b=−5b=−5, and c=−12c=−12.
Step 2: Plug the values into the quadratic formula:
x=−(−5)±(−5)2−4(4)(−12)2(4)x=2(4)−(−5)±(−5)2−4(4)(−12)
Step 3: Simplify the expression:
- Calculate the discriminant: (−5)2−4(4)(−12)=25+192=217(−5)2−4(4)(−12)=25+192=217.
- Plug it back into the equation:
x=5±2178x=85±217
Step 4: Solve for xx:
- Since 217217 is approximately 14.73, the equation becomes:
x=5±14.738x=85±14.73
This results in two possible solutions:
- x1=19.738≈2.46×1=819.73≈2.46
- x2=−9.738≈−1.22×2=8−9.73≈−1.22
Thus, the solutions are approximately x1≈2.46×1≈2.46 and x2≈−1.22×2≈−1.22.
Conclusion
It may seem very difficult to solve such quadratic equations as 4x ^ 2 – 5x – 12 = 0, however, if you follow the certain steps, it is easily solvable. In this article, we’ve explored two reliable methods: factoring and the quadratic formula As we know the quadratic formula is derived from completing the square, so we should make sure that factoring is not resulting in completing the square. Every method has its own characteristics and can be used when solving the quadratic equation depending on its nature.
FAQs About 4x ^ 2 – 5x – 12 = 0
Ans. A quadratic equation is a second-degree polynomial equation in which the highest exponent of the variable (usually xx) is 2. The general form is ax2+bx+c=0ax2+bx+c=0, where aa, bb, and cc are constants.
Ans. The solutions to the quadratic equation 4x^2 – 5x – 12 = 0 are x=−32x=−23 and x=2x=2 using the factoring method. Using the quadratic formula, the approximate solutions are x1≈2.46×1≈2.46 and x2≈−1.22×2≈−1.22.
Ans. The quadratic formula is a universal method to solve any quadratic equation, given by:x=−b±b2−4ac2ax=2a−b±b2−4acIt helps find the roots of the quadratic equation by plugging in the values of aa, bb, and cc.
Ans. To factor the equation, you find two numbers that multiply to acac (where a=4a=4 and c=−12c=−12) and add to b=−5b=−5. This leads to the factors (4x+6)(x−2)=0(4x+6)(x−2)=0, which can be solved to find x=−32x=−23 and x=2x=2.
Ans. The graph of a quadratic equation is a parabola. For the equation 4x^2 – 5x – 12 = 0, the parabola opens upwards (since a=4a=4 is positive), with a vertex representing the minimum point. The roots of the equation correspond to the x-intercepts of the parabola.
