#28. Solve the Equation with Two Square Roots

TL;DR

Solving equations with square roots by squaring both sides and checking for extraneous solutions.

Transcript

let's do problem number 28 by hand find all real solutions in exact form for the following equation so we have the square root of x plus 7 minus the square root of 2x and it's all equal to 1 we have an equation with two square roots so whenever you have an equation with two square roots you want to solve for one of the square roots and square both ... Read More

Key Insights

• ❎ Squaring both sides of an equation with square roots eliminates the square roots.
• 🫚 Checking for extraneous solutions is crucial in equations with square roots to validate the found values.
• 🧑‍🏭 Factoring quadratic equations helps find the real solutions of equations with square roots.
• 🙃 Multiplying by -1 before squaring both sides simplifies the calculation process.
• 🫚 Careful consideration and step-by-step approach are essential in solving equations with square roots.
• ❎ Understanding the principles behind squaring both sides is critical for effectively solving equations with square roots.
• 🫚 The importance of thorough checking and verifying solutions in equations with square roots cannot be overstated.

Questions & Answers

Q: How do you eliminate square roots in an equation with two square roots?

You eliminate square roots in an equation with two square roots by squaring both sides of the equation, which results in eliminating the square roots and simplifying the equation for further solution.

Q: Why is it essential to check for extraneous solutions in equations with square roots?

Checking for extraneous solutions in equations with square roots is essential to ensure that the found values are valid and satisfy the original equation, avoiding incorrect answers that may arise from squaring both sides.

Q: How do you factor a quadratic equation to find the solutions of an equation with square roots?

Factoring a quadratic equation involves finding two numbers that multiply to the constant term and add up to the coefficient of the linear term to determine the solutions of an equation with square roots.

Q: What is the significance of multiplying by -1 before squaring both sides in solving equations with square roots?

Multiplying by -1 before squaring both sides in solving equations with square roots simplifies the calculation and avoids errors by changing the signs of the equation, making the process more straightforward and efficient.

Summary & Key Takeaways

• Solving equations with square roots involves squaring both sides to eliminate the square roots.

• Check for extraneous solutions by plugging the found values back into the original equation.

• Factor the resulting quadratic equation to find the real solutions.