Final answer:
To solve the quartic equation 49x⁴ - 100 = 0, we recognize it as a difference of squares and factor it into (7x² - 10)(7x² + 10), then solve the resulting quadratic equations to find four solutions, which are the positive and negative square roots of 10/7.
Step-by-step explanation:
- The question provided is asking to solve a quartic equation of the form 49x⁴ - 100 = 0. Although the additional information provided seems to reference various quadratic equations, the actual equation in question is quartic, meaning it has a variable raised to the fourth power. To solve the equation 49x⁴ - 100 = 0, we can apply methods similar to solving quadratic equations, like factoring, completing the square, or using the quadratic formula. However, since this is a quartic equation, we would typically look to factor it into a product of quadratics or use specialized techniques for higher-order polynomials.
- Here's a step-by-step example of factoring the given quartic equation:
- Start with the equation 49x⁴ - 100 = 0.
- Notice that this is a difference of squares, as 49x⁴ is the square of (7x²) and 100 is the square of 10.
- Factor the equation as (7x² - 10)(7x² + 10) = 0.
- Solve for x by setting each factor equal to 0, leading to two separate equations: 7x² - 10 = 0 and 7x² + 10 = 0.
- Solve each quadratic equation for x. The solutions are x = ±√(10/7) and x = ±√(10/7).
- Thus, the four solutions are the positive and negative square roots of 10/7.