Perfect Square Trinomial How To Find The Value Of C

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Completing the square is a powerful technique in algebra that allows us to rewrite quadratic equations in a more convenient form. This form, often called the vertex form, reveals key information about the parabola represented by the equation, such as its vertex and axis of symmetry. The core of completing the square lies in transforming a quadratic expression into a perfect square trinomial. This article delves into the process of forming a perfect square trinomial and applies this concept to determine the value of 'c' in a given equation.

Understanding Perfect Square Trinomials

In mathematics, understanding perfect square trinomials is the key. A perfect square trinomial is a trinomial that can be factored into the square of a binomial. In simpler terms, it's a quadratic expression that results from squaring a binomial. The general form of a perfect square trinomial is:

(ax + b)^2 = a^2x^2 + 2abx + b^2$ or $(ax - b)^2 = a^2x^2 - 2abx + b^2

Notice the pattern here. The first term is the square of the first term of the binomial (a2x2a^2x^2). The last term is the square of the second term of the binomial (b2b^2). And the middle term is twice the product of the two terms of the binomial (2abx2abx or βˆ’2abx-2abx). Recognizing this pattern is crucial for completing the square.

For example, consider the trinomial $x^2 + 6x + 9$. This is a perfect square trinomial because it can be factored as $(x + 3)^2$. Here, the square root of the first term (x2x^2) is $x$, the square root of the last term (99) is $3$, and twice the product of these square roots (2βˆ—xβˆ—32 * x * 3) is equal to the middle term (6x6x).

Similarly, $x^2 - 10x + 25$ is also a perfect square trinomial, as it factors into $(x - 5)^2$. The square root of $x^2$ is $x$, the square root of $25$ is $5$, and twice the product of $x$ and $-5$ is $-10x$, matching the middle term.

However, expressions like $x^2 + 4x + 5$ are not perfect square trinomials. While the first two terms resemble the pattern, there's no integer that, when squared, equals $5$, and whose double product with $x$ gives $4x$. This highlights the importance of all three terms adhering to the perfect square trinomial pattern.

The process of completing the square relies on our ability to manipulate a quadratic expression to fit this perfect square trinomial pattern. By adding a specific constant term, we can transform a quadratic expression into a perfect square, making it easier to solve for the variable or analyze the quadratic function.

Completing the Square: The Process

Completing the square is a valuable technique used to rewrite a quadratic expression in the form $(x + a)^2 + b$ or $(x - a)^2 + b$. This form is particularly useful because it reveals the vertex of the parabola represented by the quadratic equation. The process involves manipulating the quadratic expression to create a perfect square trinomial.

Let's outline the steps involved in completing the square:

  1. Ensure the coefficient of $x^2$ is 1: If the coefficient of the $x^2$ term is not 1, divide the entire equation by that coefficient. This ensures that we can easily form the perfect square trinomial.
  2. Move the constant term to the right side of the equation: Isolate the $x^2$ and $x$ terms on one side of the equation and move the constant term to the other side. This creates space for us to add the term that will complete the square.
  3. Calculate the value to complete the square: Take half of the coefficient of the $x$ term, square it, and add this value to both sides of the equation. This is the crucial step in creating the perfect square trinomial. The value we add is always $(b/2)^2$, where $b$ is the coefficient of the $x$ term.
  4. Factor the perfect square trinomial: The left side of the equation should now be a perfect square trinomial. Factor it into the form $(x + a)^2$ or $(x - a)^2$, where $a$ is half of the coefficient of the $x$ term.
  5. Simplify the right side of the equation: Combine the constant terms on the right side of the equation.

Now, let's illustrate this process with an example. Consider the quadratic expression $x^2 + 6x + 5$. To complete the square:

  1. The coefficient of $x^2$ is already 1, so we can skip the first step.
  2. Move the constant term to the right side (conceptually; we're manipulating an expression, not solving an equation yet): $x^2 + 6x = -5$
  3. Take half of the coefficient of the $x$ term (which is 6), which is 3. Square it: $3^2 = 9$. Add 9 to both sides: $x^2 + 6x + 9 = -5 + 9$
  4. Factor the perfect square trinomial: $(x + 3)^2 = 4$

We have now completed the square. The expression $x^2 + 6x + 5$ has been rewritten as $(x + 3)^2 - 4$. This form makes it easy to identify the vertex of the parabola, which is at the point $(-3, -4)$.

Completing the square is a versatile technique with applications beyond solving quadratic equations. It's used in calculus, analytic geometry, and various other areas of mathematics. Mastering this process provides a strong foundation for understanding quadratic functions and their properties.

Applying Completing the Square to Find 'c'

Now, let's apply the process of completing the square to the given equation to find the value of 'c'. The equation is:

x^2 + 3x + c = rac{7}{4} + c

Our goal is to find the value of 'c' that will make the left side of the equation a perfect square trinomial. To do this, we'll follow the steps of completing the square:

  1. Isolate the $x^2$ and $x$ terms: In this case, the $x^2$ and $x$ terms are already isolated on the left side of the equation.

  2. Determine the value needed to complete the square: Take half of the coefficient of the $x$ term, which is 3. Half of 3 is $ rac3}{2}$. Square this value $(\frac{3{2})^2 = \frac{9}{4}$. This is the value we need to add to both sides of the equation to complete the square on the left side.

  3. Add the value to both sides: Add $ rac{9}{4}$ to both sides of the equation:

    x2+3x+c+94=74+c+94x^2 + 3x + c + \frac{9}{4} = \frac{7}{4} + c + \frac{9}{4}

  4. Focus on the left side, aiming for a perfect square trinomial: We want the left side to be a perfect square trinomial. This means that the constant term on the left side ($c + \frac{9}{4}$) should be equal to the square of half the coefficient of the $x$ term (which we already calculated as $ rac{9}{4}$).

  5. Set up an equation to solve for 'c': For the left side to be a perfect square trinomial, we need the constant term to be $ rac{9}{4}$. Therefore, we can set up the following equation:

    c=94c = \frac{9}{4}

    This ensures that the left side of the original equation becomes a perfect square trinomial: $x^2 + 3x + \frac{9}{4}$, which can be factored as $(x + \frac{3}{2})^2$

  6. Substitute the value of $c$ in the equation:

x2+3x+94=74+94x^2 + 3x + \frac{9}{4} = \frac{7}{4} + \frac{9}{4}

x2+3x+94=164x^2 + 3x + \frac{9}{4} = \frac{16}{4}

(x+32)2=4(x + \frac{3}{2})^2 = 4

Therefore, by completing the square, we have determined that the value of 'c' that makes the left side of the equation a perfect square trinomial is $ rac{9}{4}$. This allows us to rewrite the quadratic expression in a more useful form for solving equations or analyzing the function.

The Value of c

In conclusion, by carefully applying the process of completing the square, we've successfully determined the value of 'c' that transforms the given expression into a perfect square trinomial. The value of $c$ is $\frac{9}{4}$. This result highlights the power of completing the square as a technique for manipulating quadratic expressions and solving related problems. Understanding perfect square trinomials and the steps involved in completing the square is fundamental for success in algebra and beyond. This technique not only helps in solving equations but also provides valuable insights into the nature of quadratic functions and their graphical representations.