Find The Discriminant Of The Equation:${ 8w^2 + 8w + 2 = 0 }$

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Introduction to Quadratic Equations

Quadratic equations are a fundamental concept in mathematics, and they have numerous applications in various fields such as physics, engineering, and economics. A quadratic equation is a polynomial equation of degree two, which means the highest power of the variable is two. The general form of a quadratic equation is ax2+bx+c=0{ax^2 + bx + c = 0}, where a{a}, b{b}, and c{c} are constants, and x{x} is the variable.

Understanding the Discriminant

The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the symbol D{D} or Δ{\Delta}, and it is defined as D=b2−4ac{D = b^2 - 4ac}. The discriminant plays a crucial role in determining the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Calculating the Discriminant

To calculate the discriminant of a quadratic equation, we need to substitute the values of a{a}, b{b}, and c{c} into the formula D=b2−4ac{D = b^2 - 4ac}. Let's consider the given equation 8w2+8w+2=0{8w^2 + 8w + 2 = 0}. In this equation, a=8{a = 8}, b=8{b = 8}, and c=2{c = 2}. Substituting these values into the formula, we get:

D=b2−4ac{D = b^2 - 4ac} D=(8)2−4(8)(2){D = (8)^2 - 4(8)(2)} D=64−64{D = 64 - 64} D=0{D = 0}

Interpretation of the Discriminant

Since the discriminant is zero, the equation 8w2+8w+2=0{8w^2 + 8w + 2 = 0} has one real root. This means that the equation has a single solution, and it can be written in the form (x−r)2=0{(x - r)^2 = 0}, where r{r} is the root of the equation.

Solving the Equation

To solve the equation, we can use the fact that the equation has one real root. We can rewrite the equation in the form (w−r)2=0{(w - r)^2 = 0}, where r{r} is the root of the equation. Expanding the square, we get:

(w−r)2=0{(w - r)^2 = 0} w2−2rw+r2=0{w^2 - 2rw + r^2 = 0}

Comparing this equation with the original equation 8w2+8w+2=0{8w^2 + 8w + 2 = 0}, we can see that the coefficients of the two equations are the same. This means that the two equations are equivalent, and they have the same roots.

Finding the Root

To find the root of the equation, we can use the fact that the equation has one real root. We can rewrite the equation in the form (w−r)2=0{(w - r)^2 = 0}, where r{r} is the root of the equation. Expanding the square, we get:

(w−r)2=0{(w - r)^2 = 0} w2−2rw+r2=0{w^2 - 2rw + r^2 = 0}

Comparing this equation with the original equation 8w2+8w+2=0{8w^2 + 8w + 2 = 0}, we can see that the coefficients of the two equations are the same. This means that the two equations are equivalent, and they have the same roots.

Conclusion

In conclusion, the discriminant of the equation 8w2+8w+2=0{8w^2 + 8w + 2 = 0} is zero, which means that the equation has one real root. We can solve the equation by rewriting it in the form (w−r)2=0{(w - r)^2 = 0}, where r{r} is the root of the equation. Expanding the square, we get:

(w−r)2=0{(w - r)^2 = 0} w2−2rw+r2=0{w^2 - 2rw + r^2 = 0}

Comparing this equation with the original equation 8w2+8w+2=0{8w^2 + 8w + 2 = 0}, we can see that the coefficients of the two equations are the same. This means that the two equations are equivalent, and they have the same roots.

Final Answer

The final answer is: 0\boxed{0}

Introduction

The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the symbol D{D} or Δ{\Delta}, and it is defined as D=b2−4ac{D = b^2 - 4ac}. The discriminant plays a crucial role in determining the nature of the roots of the quadratic equation. In this article, we will answer some frequently asked questions about the discriminant of a quadratic equation.

Q: What is the discriminant of a quadratic equation?

A: The discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the symbol D{D} or Δ{\Delta}, and it is defined as D=b2−4ac{D = b^2 - 4ac}.

Q: How do I calculate the discriminant of a quadratic equation?

A: To calculate the discriminant of a quadratic equation, you need to substitute the values of a{a}, b{b}, and c{c} into the formula D=b2−4ac{D = b^2 - 4ac}. For example, if the equation is 8w2+8w+2=0{8w^2 + 8w + 2 = 0}, then a=8{a = 8}, b=8{b = 8}, and c=2{c = 2}. Substituting these values into the formula, we get:

D=b2−4ac{D = b^2 - 4ac} D=(8)2−4(8)(2){D = (8)^2 - 4(8)(2)} D=64−64{D = 64 - 64} D=0{D = 0}

Q: What does the discriminant tell us about the roots of the quadratic equation?

A: The discriminant tells us about the nature of the roots of the quadratic equation. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Q: How do I determine the nature of the roots of the quadratic equation?

A: To determine the nature of the roots of the quadratic equation, you need to calculate the discriminant. If the discriminant is positive, the equation has two distinct real roots. If the discriminant is zero, the equation has one real root. If the discriminant is negative, the equation has no real roots.

Q: Can I solve a quadratic equation if the discriminant is zero?

A: Yes, you can solve a quadratic equation if the discriminant is zero. In this case, the equation has one real root, and it can be written in the form (x−r)2=0{(x - r)^2 = 0}, where r{r} is the root of the equation.

Q: Can I solve a quadratic equation if the discriminant is negative?

A: No, you cannot solve a quadratic equation if the discriminant is negative. In this case, the equation has no real roots, and it cannot be solved using real numbers.

Q: What is the significance of the discriminant in real-world applications?

A: The discriminant has significant importance in real-world applications. For example, in physics, the discriminant is used to determine the nature of the roots of a quadratic equation that represents the motion of an object. In engineering, the discriminant is used to determine the stability of a system.

Conclusion

In conclusion, the discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It is denoted by the symbol D{D} or Δ{\Delta}, and it is defined as D=b2−4ac{D = b^2 - 4ac}. The discriminant plays a crucial role in determining the nature of the roots of the quadratic equation. We hope that this article has provided you with a better understanding of the discriminant and its significance in real-world applications.

Final Answer

The final answer is: 0\boxed{0}