Problem 35 - Elementary Algebra

Question

Solve. $$ 9 x 2-42 x+49=0 $$

Step-by-Step Solution

Answer

The solution is \(x = \frac{7}{3}\).

Step 1: Confirm the Equation Type

The equation given is a quadratic equation of the form \(ax^2 + bx + c = 0\) where \(a = 9\), \(b = -42\), \(c = 49\). We need to solve this equation for the variable \(x\).

Step 2: Check for Factorization

To solve the quadratic equation, let's first check if it can be factored. The expression is \(9x^2 - 42x + 49\). To factor it, we look for two numbers that multiply to \(9 \times 49 = 441\) and add up to \(-42\).

Step 3: Identify the Factors

The numbers that multiply to \(441\) and add to \(-42\) are \(-21\) and \(-21\). Hence, we can write the factors as \((3x - 7)(3x - 7) = 0\).

Step 4: Solve for x

Since \((3x - 7)(3x - 7) = 0\), we equate \(3x - 7 = 0\). Solving for \(x\), we get:\[3x - 7 = 0\] \[3x = 7\] \[x = \frac{7}{3}\].

Step 5: Verify the Solution

To ensure the solution is correct, substitute \(x = \frac{7}{3}\) back into the original equation \(9x^2 - 42x + 49\) and check if the equation holds true. It simplifies correctly to 0, so \(x = \frac{7}{3}\) is indeed the solution.

Key Concepts

Factoring Quadratics

Factoring quadratics is a strategy that can simplify solving certain kinds of quadratic equations. A quadratic equation is generally given in the form \( ax^2 + bx + c = 0 \). To factor it, we first identify two numbers that satisfy both conditions:

• Their product equals \( a \times c \).
• Their sum equals \( b \).

In the original problem, the quadratic is \( 9x^2 - 42x + 49 \). The product of the first and last coefficients, \( 9 \times 49 = 441 \), gives us the target product.Next, we find two numbers—\(-21\) and \(-21\). These multiply to \( 441 \) and add to \(-42\). Thus, the expression \( 9x^2 - 42x + 49 \) is equivalent to \((3x - 7)^2 = 0 \). Factoring quadratics by using this method transforms the problem of solving a quadratic equation into finding solutions for a product of simpler expressions, such as \( (3x - 7)(3x - 7) = 0 \). This step helps us break down complex problems into simpler parts.

Root Verification

Verifying the roots of a quadratic equation is essential to ensure that the given solution is correct. After you have factored the quadratic equation and found a potential solution for \( x \), it is a good practice to substitute that solution back into the original equation.For example, when we solve \( (3x - 7)(3x - 7) = 0 \),we find that \( x = \frac{7}{3} \) is a solution. To verify, we substitute \( x = \frac{7}{3} \) into the original equation \( 9x^2 - 42x + 49 \).By doing so, each term is simplified separately:

• 9 times the square of \( \frac{7}{3} \)
• minus 42 times \( \frac{7}{3} \)
• plus 49

All these operations should result in 0.As they accurately do, the solution is verified. This process of substituting back helps in checking that we did not miss any steps during factorization or calculation. With root verification, we confirm that the equation holds true for the value of \( x \) we have found.

Number Factoring

Number factoring is crucial when dealing with quadratic equations that require solving via factorization. It involves breaking down a composite number (a number with more than one set of divisors) into its prime factors or smaller constituent numbers that multiply to the original number. For instance, to solve \( 9x^2 - 42x + 49 \),our task involves product determination \( 9 \times 49 = 441 \).Here, the factorization is not about factoring numbers like 441 into primes, but rather finding specific whole numbers that fit both conditions of multiplication and addition from before.However, understanding prime factorization can also be helpful in simplifying complex problems. Knowing the primary divisors of a number can assist us in investigating potential pairs. In our case, exploring number pairs for 441 (which is \( 21^2 \)) helped in identifying the correct pair \(-21\), as this contributes to the factors \((3x-7)(3x-7)\).Recognizing this connection makes the factorization and, thus, the solution more straightforward, emphasizing the significance of number factoring.