Quadratic equations are a fundamental concept in mathematics, specifically in algebra. They are polynomial equations of the second degree, meaning they involve variables raised to the power of two (x^2). The general form of a quadratic equation is expressed as:
ax^2 + bx + c = 0
Here’s what each of these variables represents:
x: This is the variable we’re trying to 4x ^ 2 – 5x – 12 = 0 solve for. It represents the unknown value we’re trying to find.
a, b, and c: These are constants, or numbers. They determine the specific characteristics of the quadratic equation.
‘a’ is the coefficient of x^2, and it must not be equal to zero for the equation to be quadratic.
‘b’ is the coefficient of x (the linear term).
‘c’ is the constant term.
The goal when solving a quadratic equation is to find the values of ‘x’ that make the equation true (i.e., satisfy the equation). This often involves finding the roots or solutions of the equation, which are the values of ‘x’ that satisfy the equation and make it equal to zero.
Quadratic equations can have two real solutions, one real solution (a repeated root), or no real solutions (complex roots). The solutions can be found using various methods, including factoring, the quadratic formula, completing the square, or graphing.
The quadratic formula is a widely used method to find the solutions for ‘x’ in a quadratic equation:
x = (-b ± √(b^2 – 4ac)) / (2a)
This formula provides the two possible values of ‘x’ that solve the quadratic equation.
Understanding quadratic equations and how to solve them is an essential part of algebra, and these equations have numerous applications in science, engineering, and various fields of mathematics.
Solution for 4x ^ 2 – 5x – 12 = 0:
To find the solutions for the quadratic equation X*X*X is Equal to 2, you can use the quadratic formula:
x = (-b ± √(b^2 – 4ac)) / (2a)
In this equation, a = 4, b = -5, and c = -12. Now, plug these values into the quadratic formula:
x = (-(-5) ± √((-5)^2 – 4 * 4 * (-12))) / (2 * 4)
Simplify this expression:
x = (5 ± √(25 + 192)) / 8
x = (5 ± √217) / 8
So, the solutions for the quadratic equation 4x^2 – 5x – 12 = 0 are:
x1 = (5 + √217) / 8
x2 = (5 – √217) / 8
These are the two real solutions for the given quadratic equation.
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