Solve each equation in Exercises 47–64 by completing the square. $x^{2} + 4 x + 1 = 0$

검증된 단계별 안내

Start with the given quadratic equation: $x^2 + 4x + 1 = 0$.

Move the constant term to the right side to isolate the $x$ terms: $x^2 + 4x = -1$.

To complete the square, take half of the coefficient of $x$, which is 4, divide by 2 to get 2, then square it to get $2^2 = 4$.

Add this square (4) to both sides of the equation to maintain equality: $x^2 + 4x + 4 = -1 + 4$.

Rewrite the left side as a perfect square trinomial: $(x + 2)^2 = 3$. From here, you can proceed to solve for $x$ by taking the square root of both sides.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Completing the Square

Completing the square is a method used to solve quadratic equations by transforming the equation into a perfect square trinomial. This involves adding and subtracting a specific value to create a binomial squared, making it easier to solve for the variable.

Quadratic Equations

A quadratic equation is a second-degree polynomial equation in the form ax² + bx + c = 0. Understanding its structure is essential for applying methods like completing the square, factoring, or using the quadratic formula to find the roots.

Solving Equations Using Square Roots

Once a quadratic equation is written as a perfect square equal to a constant, you solve for the variable by taking the square root of both sides. This step introduces both positive and negative roots, which are critical for finding all solutions.

Solve each equation in Exercises 47–64 by completing the square. x2+4x+1=0x^2 + 4x + 1 = 0