Find The Discriminant Of The Equation: 2 S − 1 = − 4 S 2 2s - 1 = -4s^2 2 S − 1 = − 4 S 2 How Many Real Solutions Does The Equation Have?A. No Real SolutionsB. One Real SolutionC. Two Real Solutions

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Understanding the Quadratic Equation

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=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable. In this case, we have the equation 2s−1=−4s22s - 1 = -4s^2, which can be rewritten as 4s2+2s−1=04s^2 + 2s - 1 = 0. This is a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c = 0, where a=4a = 4, b=2b = 2, and c=−1c = -1.

The Discriminant and Its Significance

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 DD or Δ\Delta, and it is calculated using the formula D=b2−4acD = b^2 - 4ac. The discriminant is significant because it determines the nature of the solutions of the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Calculating the Discriminant

To calculate the discriminant of the given equation, we need to substitute the values of aa, bb, and cc into the formula D=b2−4acD = b^2 - 4ac. In this case, a=4a = 4, b=2b = 2, and c=−1c = -1. Substituting these values into the formula, we get:

D=(2)2−4(4)(−1)D = (2)^2 - 4(4)(-1)

D=4+16D = 4 + 16

D=20D = 20

Interpreting the Discriminant

Since the discriminant DD is positive, we can conclude that the equation has two distinct real solutions. This is because the discriminant is greater than zero, which means that the quadratic equation has two distinct roots.

Conclusion

In conclusion, the discriminant of the equation 2s−1=−4s22s - 1 = -4s^2 is D=20D = 20. Since the discriminant is positive, we can conclude that the equation has two distinct real solutions.

Real Solutions of the Equation

To find the real solutions of the equation, we need to solve the quadratic equation 4s2+2s−1=04s^2 + 2s - 1 = 0. We can use the quadratic formula to solve the equation:

s=−b±b2−4ac2as = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Substituting the values of aa, bb, and cc into the formula, we get:

s=−2±(2)2−4(4)(−1)2(4)s = \frac{-2 \pm \sqrt{(2)^2 - 4(4)(-1)}}{2(4)}

s=−2±208s = \frac{-2 \pm \sqrt{20}}{8}

s=−2±258s = \frac{-2 \pm 2\sqrt{5}}{8}

s=−1±54s = \frac{-1 \pm \sqrt{5}}{4}

Final Answer

Therefore, the final answer is:

  • A. No real solutions: This is incorrect, because the discriminant is positive.
  • B. One real solution: This is incorrect, because the discriminant is positive.
  • C. Two real solutions: This is correct, because the discriminant is positive.

The final answer is C. Two real solutions.

Understanding the Quadratic Equation

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=0ax^2 + bx + c = 0, where aa, bb, and cc are constants, and xx is the variable. In this case, we have the equation 2s−1=−4s22s - 1 = -4s^2, which can be rewritten as 4s2+2s−1=04s^2 + 2s - 1 = 0. This is a quadratic equation in the form of ax2+bx+c=0ax^2 + bx + c = 0, where a=4a = 4, b=2b = 2, and c=−1c = -1.

Q&A: Finding the Discriminant

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 DD or Δ\Delta, and it is calculated using the formula D=b2−4acD = b^2 - 4ac.

Q: Why is the discriminant important?

A: The discriminant is important because it determines the nature of the solutions of the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Q: How do I calculate the discriminant?

A: To calculate the discriminant, you need to substitute the values of aa, bb, and cc into the formula D=b2−4acD = b^2 - 4ac. For example, if you have the equation 4s2+2s−1=04s^2 + 2s - 1 = 0, you would substitute a=4a = 4, b=2b = 2, and c=−1c = -1 into the formula.

Q: What does a positive discriminant mean?

A: A positive discriminant means that the equation has two distinct real solutions. This is because the discriminant is greater than zero, which means that the quadratic equation has two distinct roots.

Q&A: Real Solutions of the Equation

Q: How do I find the real solutions of a quadratic equation?

A: To find the real solutions of a quadratic equation, you need to solve the equation using the quadratic formula. The quadratic formula is:

s=−b±b2−4ac2as = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Q: What if the discriminant is negative?

A: If the discriminant is negative, the equation has no real solutions. This is because the discriminant is less than zero, which means that the quadratic equation has no real roots.

Q: What if the discriminant is zero?

A: If the discriminant is zero, the equation has one real solution. This is because the discriminant is equal to zero, which means that the quadratic equation has one real root.

Conclusion

In conclusion, the discriminant of a quadratic equation is a value that can be calculated from the coefficients of the equation. It determines the nature of the solutions of the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.

Final Answer

Therefore, the final answer is:

  • A. No real solutions: This is incorrect, because the discriminant is positive.
  • B. One real solution: This is incorrect, because the discriminant is positive.
  • C. Two real solutions: This is correct, because the discriminant is positive.

The final answer is C. Two real solutions.

Common Quadratic Equation Mistakes

  • Mistake 1: Not calculating the discriminant correctly
    • Make sure to substitute the values of aa, bb, and cc into the formula D=b2−4acD = b^2 - 4ac.
  • Mistake 2: Not using the quadratic formula correctly
    • Make sure to use the quadratic formula s=−b±b2−4ac2as = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} to find the real solutions of the equation.
  • Mistake 3: Not checking the discriminant before solving the equation
    • Make sure to check the discriminant before solving the equation to determine the nature of the solutions.

Common Quadratic Equation Questions

  • 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 DD or Δ\Delta, and it is calculated using the formula D=b2−4acD = b^2 - 4ac.
  • Q: Why is the discriminant important?
    • A: The discriminant is important because it determines the nature of the solutions of the quadratic equation. If the discriminant is positive, the equation has two distinct real solutions. If the discriminant is zero, the equation has one real solution. If the discriminant is negative, the equation has no real solutions.
  • Q: How do I calculate the discriminant?
    • A: To calculate the discriminant, you need to substitute the values of aa, bb, and cc into the formula D=b2−4acD = b^2 - 4ac.