April 16, 2024

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=−b±b2−4ac2a

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=−(−5)±(−5)2−4(4)(−12)2(4)

X

=5±25+1928

X

=5±2178

 

Therefore, the solutions to the quadratic equation 4X^2 – 5X – 12 = 0 are:X=5+2178X=5−2178

FAQs

  1. What is a quadratic equation?
    A quadratic equation is an equation that could be written as ax^2 + bx + c = 0 when a ≠ 0.
  2. 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

  3. 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

  4. 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
  5. 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.

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