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The expression \(x^2 - 4x + 3\) is a quadratic trinomial, a fundamental type of polynomial in algebra. Understanding how to manipulate and solve equations involving this expression is crucial for various mathematical and scientific applications. We will delve into different methods for analyzing this expression, including factoring and finding its roots when set equal to zero.

Key Insights

• Factoring: The expression \(x^2 - 4x + 3\) can be factored into the product of two binomials: \((x - 1)(x - 3)\).
• Roots when equal to zero: When the expression is set to zero, forming the quadratic equation \(x^2 - 4x + 3 = 0\), the solutions (or roots) are \(x = 1\) and \(x = 3\).
• Multiple Solution Methods: This quadratic equation can be solved using factoring, the quadratic formula, or completing the square.

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Factoring the Expression \(x^2 - 4x + 3\)

Methods for Decomposition

Factoring a quadratic trinomial like \(x^2 - 4x + 3\) involves rewriting it as a product of two linear binomials. This process is often useful for simplifying expressions, solving quadratic equations, and analyzing the behavior of quadratic functions.

Factoring by Splitting the Middle Term (AC Method)

The AC method is a common technique for factoring quadratic trinomials of the form \(ax^2 + bx + c\). In the case of \(x^2 - 4x + 3\), we have \(a=1\), \(b=-4\), and \(c=3\). The method involves finding two numbers that multiply to \(a \times c\) and add up to \(b\).

For \(x^2 - 4x + 3\):

• We need two numbers that multiply to \(1 \times 3 = 3\).
• We need these same two numbers to add up to \(-4\).

The two numbers that satisfy these conditions are \(-1\) and \(-3\), since \((-1) \times (-3) = 3\) and \((-1) + (-3) = -4\).

Next, we split the middle term (\(-4x\)) using these two numbers:

x^2 - 4x + 3 = x^2 - x - 3x + 3

Now, we group the terms and factor out the greatest common factor from each group:

(x^2 - x) + (-3x + 3)

x(x - 1) - 3(x - 1)

Notice that we now have a common binomial factor of \((x - 1)\). We can factor this out:

(x - 1)(x - 3)

Thus, the factored form of \(x^2 - 4x + 3\) is \((x - 1)(x - 3)\).

Recognizing the Pattern

Sometimes, simple quadratic trinomials can be factored by recognizing a pattern. For \(x^2 - 4x + 3\), we are looking for two numbers that multiply to 3 and add to -4. As identified with the AC method, these numbers are -1 and -3. This leads directly to the factored form \((x - 1)(x - 3)\).

This method relies on understanding the relationship between the coefficients of the trinomial and the constants in the binomial factors. If a quadratic trinomial \(x^2 + bx + c\) can be factored as \((x + p)(x + q)\), then \(p \times q = c\) and \(p + q = b\).

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Solving the Quadratic Equation \(x^2 - 4x + 3 = 0\)

Finding the Roots

When the expression \(x^2 - 4x + 3\) is set equal to zero, it becomes a quadratic equation: \(x^2 - 4x + 3 = 0\). The solutions to this equation, also known as the roots or zeros, are the values of \(x\) that make the equation true. These roots represent the x-intercepts of the graph of the corresponding quadratic function \(y = x^2 - 4x + 3\).

There are several methods to solve quadratic equations. We will explore the most common ones.

Solving by Factoring

Since we have already factored the expression \(x^2 - 4x + 3\) as \((x - 1)(x - 3)\), we can use the Zero Product Property to find the roots. The Zero Product Property states that if the product of two factors is zero, then at least one of the factors must be zero.

Setting the factored form equal to zero:

\[(x - 1)(x - 3) = 0\]

Applying the Zero Product Property, we set each factor equal to zero:

\[x - 1 = 0 \quad \text{or} \quad x - 3 = 0\]

Solving each linear equation for \(x\):

\[x = 1 \quad \text{or} \quad x = 3\]

Therefore, the solutions to the equation \(x^2 - 4x + 3 = 0\) are \(x = 1\) and \(x = 3\).

Here is a video explaining how to solve a similar quadratic equation by factoring:

Video: How to Solve x^2 +4x + 3 = 0 by Factoring

Solving Using the Quadratic Formula

The quadratic formula is a general formula that can be used to solve any quadratic equation of the form \(ax^2 + bx + c = 0\). The formula is:

\[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]

For the equation \(x^2 - 4x + 3 = 0\), we have \(a=1\), \(b=-4\), and \(c=3\). Substituting these values into the quadratic formula:

\[x = \frac{-(-4) \pm \sqrt{(-4)^2 - 4(1)(3)}}{2(1)}\] \[x = \frac{4 \pm \sqrt{16 - 12}}{2}\] \[x = \frac{4 \pm \sqrt{4}}{2}\] \[x = \frac{4 \pm 2}{2}\]

This gives us two possible solutions:

\[x_1 = \frac{4 + 2}{2} = \frac{6}{2} = 3\] \[x_2 = \frac{4 - 2}{2} = \frac{2}{2} = 1\]

Again, we find the solutions are \(x = 1\) and \(x = 3\).

Solving by Completing the Square

Completing the square is another method for solving quadratic equations. It involves rewriting the quadratic expression in the form \((x - h)^2 = k\) and then taking the square root of both sides.

Starting with the equation \(x^2 - 4x + 3 = 0\):

Move the constant term to the right side of the equation:

\[x^2 - 4x = -3\]

To complete the square on the left side, take half of the coefficient of the \(x\) term (\(-4\)), square it \(( (-4)/2 )^2 = (-2)^2 = 4\), and add it to both sides of the equation:

\[x^2 - 4x + 4 = -3 + 4\] \[x^2 - 4x + 4 = 1\]

The left side is now a perfect square trinomial, which can be factored as \((x - 2)^2\):

\[(x - 2)^2 = 1\]

Take the square root of both sides:

\[\sqrt{(x - 2)^2} = \pm \sqrt{1}\] \[x - 2 = \pm 1\]

Solve for \(x\) by adding 2 to both sides:

\[x = 2 \pm 1\]

This gives us the two solutions:

\[x_1 = 2 + 1 = 3\] \[x_2 = 2 - 1 = 1\]

Once more, the solutions are \(x = 1\) and \(x = 3\).

Here's a video demonstrating the completing the square method for a similar equation:

Video: Solving Quadratic Equations by Completing The Square: x²-4x+3=0

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Summary of Solution Methods for \(x^2 - 4x + 3 = 0\)

The following table summarizes the results obtained using different methods to solve the equation \(x^2 - 4x + 3 = 0\):

Method	Steps	Solutions
Factoring	Factor the trinomial into \((x-1)(x-3)\) and set each factor to zero.	\(x=1, x=3\)
Quadratic Formula	Substitute \(a=1\), \(b=-4\), \(c=3\) into \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\).	\(x=1, x=3\)
Completing the Square	Rewrite the equation as \((x-2)^2 = 1\) and take the square root of both sides.	\(x=1, x=3\)

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The Graph of the Quadratic Function \(y = x^2 - 4x + 3\)

Understanding the Parabola

The expression \(x^2 - 4x + 3\) represents a quadratic function when set equal to \(y\), i.e., \(y = x^2 - 4x + 3\). The graph of a quadratic function is a parabola. For a quadratic function in the form \(y = ax^2 + bx + c\), if \(a > 0\), the parabola opens upwards, and if \(a < 0\), it opens downwards.

In \(y = x^2 - 4x + 3\), \(a=1\), which is greater than 0, so the parabola opens upwards. The roots of the equation \(x^2 - 4x + 3 = 0\) correspond to the x-intercepts of the parabola, which are the points where the parabola crosses the x-axis. Since the roots are \(x=1\) and \(x=3\), the parabola intersects the x-axis at \((1, 0)\) and \((3, 0)\).

The vertex of the parabola is the minimum or maximum point. For a parabola opening upwards (\(a>0\)), the vertex is the minimum point. The x-coordinate of the vertex is given by \(x = \frac{-b}{2a}\).

For \(y = x^2 - 4x + 3\):

\[x = \frac{-(-4)}{2(1)} = \frac{4}{2} = 2\]

To find the y-coordinate of the vertex, substitute \(x = 2\) back into the equation \(y = x^2 - 4x + 3\):

\[y = (2)^2 - 4(2) + 3\] \[y = 4 - 8 + 3\] \[y = -1\]

The vertex of the parabola is \((2, -1)\).

Parabolas are found in many real-world applications, such as the trajectory of a projectile, the shape of satellite dishes, and the design of bridges.

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FAQ

What is a quadratic expression?

A quadratic expression is a polynomial of degree 2, meaning the highest power of the variable is 2. It is typically written in the form \(ax^2 + bx + c\), where \(a\), \(b\), and \(c\) are constants and \(a \neq 0\). In the expression \(x^2 - 4x + 3\), \(a=1\), \(b=-4\), and \(c=3\).

What are the roots of a quadratic equation?

The roots of a quadratic equation \(ax^2 + bx + c = 0\) are the values of \(x\) that satisfy the equation. Graphically, these are the x-intercepts of the parabola represented by the corresponding quadratic function.

Can all quadratic expressions be factored?

Not all quadratic expressions with integer coefficients can be factored into two linear binomials with integer coefficients. However, all quadratic equations with real coefficients have solutions (real or complex) that can be found using the quadratic formula.

How can I check if my factored expression is correct?

You can check your factored expression by multiplying the binomials back together using the FOIL method (First, Outer, Inner, Last). If the result is the original quadratic expression, then your factoring is correct. For example, \((x - 1)(x - 3) = x \times x + x \times (-3) + (-1) \times x + (-1) \times (-3) = x^2 - 3x - x + 3 = x^2 - 4x + 3\).

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References