Problem 55
Question
Solve the equation by factoring. Use a graphing calculator to check your solution if you wish. $$ 6 b^{2}-72 b+216=0 $$
Step-by-Step Solution
Verified Answer
The solution to the equation \(6b^2 - 72b + 216 = 0\) is \(b = 6\).
1Step 1: Set up the quadratic equation
The equation is already in the standard form of a quadratic equation \( ax^2 + bx + c = 0 \) as \(6b^2 - 72b + 216 = 0\).
2Step 2: Simplify the equation through factoring
First, factor out the greatest common factor (GCF). The GCF in this case is 6, and the equation simplifies to \(6(b^2 - 12b + 36) = 0\). Then, you have to isolate the brackets and solve the quadratic equation left. This equation is a perfect square trinomial and it can be factored as \(6(b - 6)^2 = 0\).
3Step 3: Solve for b
To find the solutions for this equation, set the factored equation equal to zero. This leads to \(b - 6 = 0\). Solving for \(b\) gives the final answer of \(b = 6\) .
Key Concepts
Greatest Common FactorQuadratic FormulaPerfect Square Trinomial
Greatest Common Factor
When you encounter a quadratic equation, the first step in solving it through factoring is often to identify and factor out the greatest common factor (GCF) from all the terms. The GCF is the largest number or expression that can evenly divide each of the terms in the equation. In this exercise, the equation given is \(6b^2 - 72b + 216 = 0\).
Upon inspection, we need to find the GCF of the coefficients 6, 72, and 216. The highest number that can accurately divide each of these is 6.Once identified, you factor it out of the equation. This simplifies the equation to \(6(b^2 - 12b + 36) = 0\). By removing the GCF, the process of factoring becomes more straightforward, making it easier to identify other characteristics of the equation, like whether it is a perfect square trinomial.
Upon inspection, we need to find the GCF of the coefficients 6, 72, and 216. The highest number that can accurately divide each of these is 6.Once identified, you factor it out of the equation. This simplifies the equation to \(6(b^2 - 12b + 36) = 0\). By removing the GCF, the process of factoring becomes more straightforward, making it easier to identify other characteristics of the equation, like whether it is a perfect square trinomial.
Quadratic Formula
The quadratic formula is an essential tool used for solving quadratic equations when they cannot be easily factored. It's defined as:\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]This formula works with all quadratic equations of the form \( ax^2 + bx + c = 0 \).
In the exercise example, we simplified the equation to \(b^2 - 12b + 36 = 0\) after factoring out the GCF. While this specific equation can be factored by other means (as a perfect square trinomial), the quadratic formula offers a foolproof approach.By plugging the coefficients a, b, and c from the equation into the quadratic formula, you can find all possible solutions for \(b\). This formula is especially useful when factoring seems complicated or when you're unable to recognize any patterns.
In the exercise example, we simplified the equation to \(b^2 - 12b + 36 = 0\) after factoring out the GCF. While this specific equation can be factored by other means (as a perfect square trinomial), the quadratic formula offers a foolproof approach.By plugging the coefficients a, b, and c from the equation into the quadratic formula, you can find all possible solutions for \(b\). This formula is especially useful when factoring seems complicated or when you're unable to recognize any patterns.
Perfect Square Trinomial
A perfect square trinomial is a special kind of quadratic equation that can be expressed as the square of a binomial. This means that the equation can be written in the form \((x + d)^2\) or \((x - d)^2\), making it straightforward to solve once factored.
In the exercise, the equation \(b^2 - 12b + 36\) appears after factoring out the GCF. You can recognize it as a perfect square trinomial because it fits the pattern where the first and last terms are perfect squares while the middle term corresponds to twice the product of the square roots of the first and last terms (i.e., \(2 \times b \times 6 = 12b\)).Thus, \(b^2 - 12b + 36\) factors as \((b - 6)^2\). Once identified and rewritten, you solve for \(b\) by setting \((b - 6)^2 = 0\). This gives a clear solution: \(b = 6\). Understanding this pattern can save time and simplify future factoring problems.
In the exercise, the equation \(b^2 - 12b + 36\) appears after factoring out the GCF. You can recognize it as a perfect square trinomial because it fits the pattern where the first and last terms are perfect squares while the middle term corresponds to twice the product of the square roots of the first and last terms (i.e., \(2 \times b \times 6 = 12b\)).Thus, \(b^2 - 12b + 36\) factors as \((b - 6)^2\). Once identified and rewritten, you solve for \(b\) by setting \((b - 6)^2 = 0\). This gives a clear solution: \(b = 6\). Understanding this pattern can save time and simplify future factoring problems.
Other exercises in this chapter
Problem 55
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Use the quadratic formula or factoring to find the roots of the polynomial. Write your solutions in simplest form. \(6 x^{2}-2 x-7=0\)
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