Welcome to our blog, we discuss how to solve quadratic equations? Specifically, we will focus on solving the quadratic equation 4x^2 – 5x – 12 = 0. Quadratic equations are an important part of algebra and it is also used in many fields of mathematics and science. In this article, we discuss step by step solution to solve this equation.
What are Quadratic Equations?
Quadratic equation is an essential part of algebra and mathematics. In equation we always use form of ax^2 + bx + c = 0, where a, b, c are constants and x is a variable representing an unknown value. In quadratic equation highest power of the variable x is, linear equations power of the variable x is 1 and cubic equations power of the variable x is 3.
Now we represent general form of a quadratic equation:
ax^2 + bx + c = 0
where:
- a represents coefficient of the quadratic term (x^2)
- b represents coefficient of the linear term (x)
- c represents the constant term
Quadratic equations always have many types of solutions based on the discriminant (b^2 – 4ac) within the quadratic formula.
Now the discriminant determines the nature of the roots as:
- If the discriminant is positive (b^2 – 4ac > 0), then quadratic equation has two distinct real roots and x-axis at two points.
- If the discriminant is zero (b^2 – 4ac = 0), then quadratic equation has one real root and x-axis at one point.
- If the discriminant is negative (b^2 – 4ac < 0), then quadratic equation has no real roots and does not intersect the x-axis at all.
Quadratic equations are used in many fields like engineering, finance, physics and computer graphics. It is important to solve problems and finding the maximum or minimum values of functions.
Steps by step solution of Quadratic Equations:
The equation 4x^2 – 5x – 12 = 0 is a quadratic equation, where the highest value of the variable x is 2. The form of quadratic equations is ax^2 + bx + c = 0, where a, b, c are constants and a=4, b=-5 and c=-12
Now, we can use quadratic formula to solve the quadratic equation:
x = (-b ± √(b^2 – 4ac)) / 2a
Put the values of a, b and c into the formula,
Now x= (5 ± √((-5)^2 – 4 * 4 * -12)) / 2 * 4
then, solve the equation:
x = (5 ± √(25 + 192)) / 8
x = (5 ± √217) / 8
Now we got two possible solutions are:
x = (5 + √217) / 8
x = (5 – √217) / 8
Finally, we found quadratic equation 4x^2 – 5x – 12 = 0 two real solutions is:
x =(5-√217)/8=-1.216
x =(5+√217)/8= 2.466
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Conclusion:
Finally, we solve quadratic equations with formula. these equations used to solve the problems of mathematical and practical. In this article we solve quadratic equation 4x^2 – 5x – 12 = 0 by using the quadratic formula and get two real solutions as (5 + √217) / 8 and (5 – √217) / 8.
