### Solving Quadratic Equations Using the Quadratic Formula

To solve the quadratic equation 4X^2 – 5X – 12 = 0 using the quadratic formula, we need to identify the values of a, b, and c in the equation.

The quadratic formula is:$x$

$=ab±b−ac $In the equation 4X^2 – 5X – 12 = 0, a = 4, b = -5, and c = -12.

We can substitute these values into the quadratic formula and simplify:$X$

$=()()±()2−()() $X

$X$

$=8±+ $X

$X$

$=8±217$

Therefore, the solutions to the quadratic equation 4X^2 – 5X – 12 = 0 are:$X=8+217 $$X=8−217 $

### FAQs

- What is a quadratic equation?

A quadratic equation is an equation that could be written as ax^2 + bx + c = 0 when a ≠ 0. - What are the three main ways to solve quadratic equations?

The three main ways to solve quadratic equations are factoring, using the quadratic formula, and completing the square - What is the quadratic formula?

The quadratic formula is a formula that can be used to solve any quadratic equation. It is: x = (-b ± √(b^2 – 4ac)) / 2a - What is the discriminant?

The discriminant is the part of the quadratic formula under the square root sign, b^2 – 4ac. It can be used to determine the number and type of solutions to a quadratic equation - How do I know which method to use to solve a quadratic equation?

There is no one-size-fits-all answer to this question. However, some general guidelines are: try factoring first, then use the quadratic formula if factoring doesn’t work, and use completing the square as a last resort

### Solve: 4x ^ 2 – 5x – 12 = 0

To solve the equation 4x^2 – 5x – 12 = 0, we can use the quadratic formula, which gives:

x = (-b ± sqrt(b^2 – 4ac)) / 2a

Here, a = 4, b = -5, and c = -12, so substituting these values, we get:

x = (-(-5) ± sqrt((-5)^2 – 4(4)(-12))) / 2(4)

Simplifying the expression under the square root, we get:

x = (5 ± sqrt(25 + 192)) / 8

x = (5 ± sqrt(217)) / 8

Therefore, the solutions to the equation 4x^2 – 5x – 12 = 0 are:

x = (5 + sqrt(217)) / 8

x = (5 – sqrt(217)) / 8

So, x can be approximated to:

x ≈ 2.25 or x ≈ -1.75

Hence, the solutions to the equation are x ≈ 2.25 and x ≈ -1.75.

### Solve: x + 9 = 18 + -2x

To solve the equation x + 9 = 18 – 2x, we can first simplify it by adding 2x to both sides, which gives:

x + 2x + 9 = 18

Combining like terms, we get:

3x + 9 = 18

Next, we can isolate x on one side by subtracting 9 from both sides, which gives:

3x = 9

Finally, we can solve for x by dividing both sides by 3, which gives:

x = 3

Therefore, the solution to the equation x + 9 = 18 – 2x is x = 3.