Are you ready to delve into the intriguing world of mathematical equations? Today, we will explore a fascinating quadratic equation. In this article, we will unravel the mystery surrounding the equation X2 + Bx + 16 = 0 and discover its only solution, which is X = 4. However, one crucial piece of this puzzle remains unknown: the value of B. Join us on this mathematical journey as we unearth the value that completes this equation!
Before we embark on our quest to find the value of B, let’s review the basics. A quadratic equation is a second-degree polynomial equation, often represented in the form ax2 + bx + c = 0. Our equation, X2 + Bx + 16 = 0, falls under this category. It is significant to note that this equation possesses a unique solution, with X equaling 4. Now, the mystery lies in determining the value of B that makes this equation hold true.
Understanding Quadratic Equations
What is a Quadratic Equation?
A quadratic equation is a polynomial equation of degree 2, which means its highest power term is squared. It can be written in the form ax^2 + bx + c = 0, where a, b, and c are constants and x represents the variable. Solving a quadratic equation means determining the values of x that satisfy the equation and make it true.
The General Formula for Quadratic Equations
In order to solve a quadratic equation, one can apply either factoring, completing the square, or using the quadratic formula. The quadratic formula is particularly useful since it can be applied to any quadratic equation. For a quadratic equation in the form ax^2 + bx + c = 0, the quadratic formula states that the values of x can be found using the formula x = (-b ± √(b^2 – 4ac)) / (2a).
The Only Solution of the Equation x^2 + bx + 16 = 0 is x = 4
Interpreting the Given Information
The given information states that the only solution of the equation x^2 + bx + 16 = 0 is x = 4. This essentially means that when the value of x is 4, the equation is satisfied. To find the value of b, we need to substitute x = 4 into the equation and solve for b.
Solving the Equation for b
Substituting x = 4 into the equation x^2 + bx + 16 = 0, we get 4^2 + 4b + 16 = 0. Simplifying further, we have 16 + 4b + 16 = 0 which becomes 4b + 32 = 0 after combining like terms. Now, subtracting 32 from both sides of the equation gives us 4b = -32. Finally, dividing both sides by 4 yields the solution b = -8.
Understanding the Nature of the Quadratic Equation
The Discriminant in Quadratic Equations
The discriminant is a mathematical term used to determine the nature of the solutions of a quadratic equation. It is denoted by the symbol Δ and is calculated using the formula Δ = b^2 – 4ac. By analyzing the value of the discriminant, one can determine if the quadratic equation has real solutions, imaginary solutions, or equal solutions.
Analyzing the Quadratic Equation x^2 + bx + 16 = 0
In the given equation x^2 + bx + 16 = 0, the coefficients a, b, and c are 1, b, and 16 respectively. Therefore, when we calculate the discriminant Δ = b^2 – 4ac, we get Δ = b^2 – 4(1)(16) which simplifies to Δ = b^2 – 64.
Deducing the Value of b from the Given Information
Using the Discriminant to Find b
From the given information that the only solution of x^2 + bx + 16 = 0 is x = 4, we can deduce that the discriminant Δ must equal zero. Since Δ = b^2 – 64, setting Δ to zero gives us the equation b^2 – 64 = 0.
Solving the Equation for b
To find the value of b, we need to solve the equation b^2 – 64 = 0. Taking the square root of both sides, we obtain b = ± √64. Therefore, the values of b can be either 8 or -8, depending on the positive or negative square root.
Applying the Given Value
Substituting b = 8 into the Equation
If we substitute b = 8 into the equation x^2 + bx + 16 = 0, we get x^2 + 8x + 16 = 0. Solving this quadratic equation will yield different solutions for x compared to when b = -8. Therefore, the given value of b = 8 cannot satisfy the condition x = 4 as the only solution.
Substituting b = -8 into the Equation
If we substitute b = -8 into the equation x^2 + bx + 16 = 0, we get x^2 – 8x + 16 = 0. Solving this quadratic equation will lead to x = 4 as the only solution. Hence, when b = -8, the equation’s only solution is x = 4.
1. What is the equation given?
The equation given is X^2 + Bx + 16 = 0.
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2. What is the solution to the equation?
The solution to the equation is X = 4.
3. What is the purpose of finding the value of B?
The purpose of finding the value of B is to determine the constant term in the equation.
4. How can we find the value of B?
To find the value of B, we can substitute the value of X (which is 4) into the equation and solve for B.
5. What is the value of B?
By substituting X = 4 into the equation X^2 + Bx + 16 = 0 and solving for B, we find that the value of B is -8.
Conclusion:
The value of B in the equation X^2 + Bx + 16 = 0, where the only solution is X = 4, is -8.