Which Statement is True About the Quadratic Equation 8×2 − 5x + 3 = 0?
When it comes to the quadratic equation 8x^2 – 5x + 3 = 0, there are a few statements that can be evaluated to determine their truthfulness. One statement could be whether or not the equation has real solutions. Another statement might question if the leading coefficient is positive or negative. Let’s take a closer look at these statements and uncover which one holds true for this particular quadratic equation.
To begin with, we need to determine if the given quadratic equation has real solutions. This can be done by analyzing the discriminant, which is calculated using the formula b^2 – 4ac. In our case, where a = 8, b = -5, and c = 3, plugging these values into the discriminant formula gives us (-5)^2 – 4(8)(3) = 25 – 96 = -71. Since the discriminant is negative (-71), we can conclude that there are no real solutions to this quadratic equation.
Moving on to another statement about this equation: Is the leading coefficient positive or negative? The leading coefficient refers to the coefficient of x^2 term in a quadratic equation. Here, our leading coefficient is positive (8x^2), as it is greater than zero. Therefore, we can confidently state that for the quadratic equation 8x^2 – 5x + 3 = 0, the statement “the leading coefficient is positive” holds true.
Understanding the Quadratic Equation
How to Solve Quadratic Equations
When it comes to solving quadratic equations, it’s important to have a clear understanding of the steps involved. The quadratic equation is in the form of ax^2 + bx + c = 0, where a, b, and c are constants. To solve such an equation, we can employ various methods, including factoring, completing the square, or using the quadratic formula.
Factoring: If possible, we can factorize the quadratic equation by finding two binomials whose product equals zero. For example:
x^2 + 5x + 6 = 0
- (x + 3)(x + 2) = 0
In this case, x = -3 or x = -2 are the solutions.
Completing the Square: This method involves transforming the given equation into a perfect square trinomial and then isolating x. Here’s an example:
x^2 – 4x – 12 = 0
(x – 2)^2 -16 = 0
- (x – 2)^2 =16
Taking both positive and negative square roots gives us x=6 or x=-10 as solutions.
- Quadratic Formula: The most widely used method for solving quadratic equations is by employing the quadratic formula:
x = (-b ± √(b^2-4ac)) / (2a)
By substituting the values of a,b,c from our given equation into this formula and simplifying further, we can find the solutions for x.
The Components of the Quadratic Equation
To better understand how a quadratic equation works and its significance in mathematics and real-world applications, let’s break down its components:
- Coefficients: The coefficients a, b, and c represent the numbers multiplying the different powers of x in the equation.
- Variable: The variable x represents an unknown value or values that we are trying to solve for.
- Constants: The constants in a quadratic equation are represented by numbers without variables attached to them.
- Discriminant (b^2-4ac): The discriminant is found within the quadratic formula and helps determine the nature of solutions possible for a given quadratic equation.
Understanding these key components will enable you to analyze and solve quadratic equations more effectively in various mathematical scenarios and practical applications.
Using the Quadratic Formula to Find Solutions
Let’s dive into using the quadratic formula to find solutions for the given equation, 8x^2 – 5x + 3 = 0. The quadratic formula is a powerful tool that allows us to solve any quadratic equation of the form ax^2 + bx + c = 0, where a, b, and c are coefficients.
The quadratic formula is:
x = (-b ± √(b^2 – 4ac)) / (2a)
In our case, a = 8, b = -5, and c = 3. Let’s plug these values into the formula and simplify it step by step:
Step 1: Calculate discriminant (b^2 – 4ac):
discriminant = (-5)^2 – (4 * 8 * 3)
= 25 -96
Since the discriminant is negative (-71), we can conclude that there are no real solutions for this equation. Instead, we have two complex solutions.
Step 2: Apply the quadratic formula:
x_1 = (-(-5) + √(-71))/(2*8)
= (5i + √71i)/(16)
x_2 = (-(-5) – √(-71))/(2*8)
= (5i – √71i)/(16)
Here, i represents the imaginary unit (√-1).
- The discriminant is negative (-71), indicating no real solutions.
- The two complex solutions for this equation are x_1=(5i+√71i)/16 and x_2=(5i-√71i)/16.
It’s important to note that while there may not be real solutions in some cases, complex solutions can still provide valuable insights and mathematical understanding.