Problem 74 - Algebra 2

Question

Solve each equation. $$ x^{4}-14 x^{2}+49=0 $$

Step-by-Step Solution

Answer

The solutions for the equation are x = \(\sqrt{7}\) and x = -\(\sqrt{7}\).

Step 1: Use Substitution Method

Substitute \(X=x^2\). The equation becomes as follows: \(X^2 - 14X + 49 = 0\).

Step 2: Factor the Quadratic Equation

Factorize the equation to get \((X-7)^2 = 0\).

Step 3: Solve for X

When \((X - 7)^2 = 0\), X can be 7.

Step 4: Replace X with \(x^2\)

Now replace X with \(x^2\), so the equation becomes \(x^2=7\).

Step 5: Solve for x

To solve for x, take the square root of both sides. Remembering there are two possible answers, positive and negative, the solutions for x are \(\sqrt{7}\) and -\(\sqrt{7}\).

Key Concepts

Substitution Method

The substitution method is a helpful technique for solving complex polynomial equations by simplifying them into familiar forms. It's particularly useful when dealing with quadratic equations within higher-power equations. This approach involves the following:

• Identifying a segment of the equation that forms a simpler pattern, like a square or a cube.
• Substituting this segment with a new variable, making the equation easier to manage.

In our problem, we identified that replacing the segment \(x^2\) with \(X\) simplified the original equation. So, \(X = x^2\) transformed the equation \(x^4 - 14x^2 + 49 = 0\) into a quadratic form \(X^2 - 14X + 49 = 0\). This maneuver not only simplifies the computation but empowers us to use straightforward techniques, such as factoring, to solve equations.

Factoring Quadratic

Factoring is a method of breaking down an expression into products of simpler factors. To factor a quadratic equation appropriately, we need to look for two numbers that multiply to the constant term and add up to the coefficient of the linear term. Here's a breakdown:

• For the equation \(X^2 - 14X + 49 = 0\), 7 and 7 are the numbers that multiply to 49 and add to 14.
• This allows us to express the equation as \((X-7)(X-7) = 0\), or \((X-7)^2 = 0\).

This decomposition simplifies the equation significantly. After factoring, solving the quadratic becomes a straightforward task. We find that \(X\) equals 7 by setting the factor expression to zero. Factoring quadratics not only makes the equation easy to solve but also unveils deeper insights into how terms interact with each other.

Solving Polynomials

Solving polynomials involves finding the values of the variable that make the polynomial equal zero. When solving for the original variable \(x\) in our substituted equation, there's a last step involved. Here’s how it was achieved:

• From the factored form \((X-7)^2 = 0\), we knew \(X = 7\).
• Since we previously substituted \(X = x^2\), we now return to the original variable by setting \(x^2 = 7\).

To find \(x\), we take the square root of both sides. This yields two solutions, \(x = \sqrt{7}\) and \(x = -\sqrt{7}\). This step highlights an important concept of solving polynomials wherein each root must be considered, be it a positive or negative outcome, reflecting the acknowledgment of symmetry in quadratic equations. By understanding these roots, we solve the polynomial equation entirely and reflect the possible states of its variables.