**Method Solution: 4x ^ 2 – 5x – 12 = 0**

**STEP 1: Equation at step’s conclusion is (22×2 – 5x) – 12 = 0.****Step 2: Dividing the middle word in an attempt to factor**

**2.1 Factorization 4x ^ 2 – 5x – 12 = 0**

- The coefficient of the first term, 4×2, is 4.
- The coefficient of the middle word, -5x, is -5.
- Finally, “the constant” is -12.

**Step 1:**Divide the first term’s coefficient by the constant**. 4 * -12 = -48****Step 2:**Locate two -48 factors so that their sum equals the middle term’s coefficient, which is

-5.-48} adding {1} equals

- ~-47~ ~-24~ + ~2~ equals ~-22~
- ~-8~ + ~6~ equals ~-2~
- ~-6~ + ~8~ equals ~2~
- ~-4~ + ~12~ equals ~8~ ~-3~ adding ~16~ equals ~13~ ~-2~ + ~24~ equals ~22} ~-1}

**Observation: No two of these criteria are present**

**Conclusion:** Factoring a trinomial is impossible.

The equation at the conclusion of step 2 is 4x ^ 2 – 5x – 12 = 0.

**Vertex Finding for a Parabola**

3.1 Determine the Vertex for y = **4x ^ 2 – 5x – 12 = 0**

- The vertex of a parabola is its highest or lowest point. As a result of opening up, our parabola has an absolute minimum, or lowest point. Because the coefficient of the first term, 4, is positive (greater than zero), we already knew this before plotting “y”.
- A vertical line of symmetry travels across the vertex of every parabola. Due to its symmetry, the line of symmetry would, for instance, cross the middle of the parabola’s two x-intercepts, which are its roots or solutions. That is, presuming the parabola has two real solutions.
- Parabolas can be used to mimic a wide range of real-world situations, such as the height above the ground of an object thrown upward after a predetermined amount of time.
- We can get information from the parabola’s vertex, such as the highest point an object thrown upwards can go. Thus, we must be able to ascertain the coordinates of the vertex.
- -B/(2A) gives the x-coordinate of the vertex for any parabola, Ax2+Bx+C. The x coordinate in our instance is 0.6250.
- We may determine the y-coordinate by plugging the parabola formula, 0.6250, for x: y = 4.0 * 0.62 * 0.62 – 5.0 * 0.62 – 12.0 or y = -13.562

**The quadratic equation can be solved by completing the square (3.2).**

- This method can also be used to solve the equation 4x ^ 2 – 5x – 12 = 0.
- To find the coefficient of the first term, divide both sides of the equation by 4: x2-(5/4)x-3 = 0
- To both sides of the equation, add 3:

**x2-(5/4)x equals 3.**

Here’s the cunning part: Once you take the coefficient of x, which is 5/4, divide it by two to get 5/8, and then square it to get 25/64.

**Add 25/64 to both sides of the equation:**

- We have the following on the right side: 3 + 25/64, or (3/1) + (25/64)
- The two fractions’ common denominator is 64. The sum of (192/64) and (25/64) is 217/64.

## Thus, After Adding to Both Sides, We Finally Get

**(217/64) = x2-(5/4)x+(25/64)**

The left hand side has been finished into a perfect square by adding 25/64:

** (x-(5/8)) equals x2-(5/4)xadding 25/64) multiplying it by x – (5/8)) equals amount to x – (5/8))**

Two things are equal to one other if they maybe are equal to the same thing.

**Since (x2-(5/4)x adding 25/64) equals (x-(5/8)) or****(x2-(5/4)x+(25/64) equals 217/642 after it, through the transitivity law, (x-(5/8))2 equals 217/64****This equating will be stated as Eq. #3.2.1.**

- Stated to the square root rule, two valued have equal square roots when they are value equals.
- Take note that (x-(5/8)) square root2 is equal to (x-(5/8)).2/2 is equal to x-(5/8)1 equals x-(5/8).
- Now, using Eq. #3.2.1 and the Square Root Principle, we obtain: x-(5/8) = √ 217/64.
- To get: Add 5/8 to each side.
- x is equal to 5/8 + – 217/64.

**Given it a square root has two things a positive thing and a negative thing**

x2 – (5/4)There are two methods to answer x – 3 = 0:

- x equals √ 217/64 adding 5/8 or x equals √ 217/64 – 5/8
- Keep in brain that √ 217/64 can also be stated as √ 217 / √ 64, and √ 217 / 8.

**Utilizing the Quadratic Formula, solve the quadratic equation 3.3**.

4x ^ 2 – 5x – 12 = 0 can be solved using the quadratic formula.

The solution to the equation Ax2+Bx+C = 0, where A, B, and C are numbers—often referred to as coefficients—is provided by the quadratic formula, x.

– B ± √ B2-4AC x = ————————

In this instance, A = 4 B = -5 C = -12

Thus, B2 – 4AC = 25 – (-192) = 217

Making use of the quadratic formula

Five plus or minus two hundred seventeen x equals —————

√ 217 equals 14.7309 when rounded to four decimal places.

Thus, the equation we are examining now is x = ( 5 ± 14.731 ) / 8.

**Two practical answers**

x = (1/217+5) = 2.466

or

(5-217)/8 = -1.216 for x

There were two answers discovered:

x = (5+√217)/8= 2.466 x = (5-√217)/8= -1.216

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