Problem 12 - Algebra 2

Question

Solve each equation by factoring. Then graph. \(x^{2}-3 x=-\frac{9}{4}\)

Step-by-Step Solution

Answer

The solution is \( x = \frac{3}{2} \). The graph is a parabola tangent to the x-axis at \( x = \frac{3}{2} \).

Step 1: Move All Terms to One Side

First, move all terms to one side of the equation so it is set to zero. Start with the equation: \[ x^2 - 3x = -\frac{9}{4} \] Add \( \frac{9}{4} \) to both sides:\[ x^2 - 3x + \frac{9}{4} = 0 \]

Step 2: Eliminate Fractions

To remove the fraction, multiply every term by 4 to clear the denominator:\[ 4(x^2 - 3x + \frac{9}{4}) = 0 \] Simplify:\[ 4x^2 - 12x + 9 = 0 \]

Step 3: Check for Factoring

Check if the quadratic \( 4x^2 - 12x + 9 \) can be factored. It can be written as: \[ (2x - 3)(2x - 3) = 0 \] Thus, the equation is a perfect square.

Step 4: Solve the Factored Equation

Set the factored form equal to zero:\[ (2x - 3)(2x - 3) = 0 \] This implies:\[ 2x - 3 = 0 \] Solve for \( x \) by adding 3 and dividing by 2:\[ 2x = 3 \] \[ x = \frac{3}{2} \] This equation has a repeated root, \( x = \frac{3}{2} \).

Step 5: Graph the Solution

Since the equation is a perfect square, the graph is a parabola that touches the x-axis at its vertex. The vertex is at \( x = \frac{3}{2} \). Draw the parabola that opens upward and is tangent to the x-axis at \( x = \frac{3}{2} \). The vertex form of the parabola is given by:\[ y = (2x - 3)^2 \]Plot this on the coordinate plane, showing only the tangent point on the x-axis.

Key Concepts

Factoring Quadratics

Factoring quadratics is a method used to solve quadratic equations by expressing them as a product of simpler binomials. This technique is particularly handy when dealing with trinomials, like in our original equation. Once you set the equation equal to zero, the next step is to determine whether it can be factored into two binomial expressions.

In the case of a perfect square trinomial, you end up with a factor that repeats. For example, if you factor the quadratic equation \( 4x^2 - 12x + 9 \), you find:

• It simplifies to \( (2x - 3)(2x - 3) \).
• This repetition indicates that the quadratic has a repeated root, meaning it crosses the x-axis at a single point.

When solving, set each binomial factor to zero to find the value of \( x \). If they are the same, this confirms the repeated root.
Factoring quadratics makes it easier to identify critical points in graphing, namely, the roots of the equation.

Vertex Form

The vertex form of a quadratic equation provides a clear insight into the geometry of a parabola, especially its vertex and axis of symmetry. Often written as \( y = a(x - h)^2 + k \), this form makes it easy to identify the vertex of a parabola at point \( (h, k) \).

For the quadratic in the example, we ultimately reformulate it to show its vertex form, \[ y = (2x - 3)^2 \], because:

• This expression represents a perfect square trinomial, suggesting a vertex formed at the point where \( 2x - 3 = 0 \).
• Solving gives the vertex \( x = \frac{3}{2} \).

The vertex form is particularly useful when graphing, as it provides both a starting point (the vertex) and the direction in which the parabola opens (upward for a positive \( a \), downward for negative \( a \)).
Understanding how to convert a standard quadratic equation into vertex form is an essential skill to accurately visualizing its graph.

Parabola Graphing

Graphing a parabola is straightforward once you understand the components such as the vertex, axis of symmetry, and direction of opening. After factoring or rewriting the quadratic equation into vertex form, these elements guide you in sketching the curve.

For the example equation, \( y = (2x - 3)^2 \), we know:

• The vertex is \( x = \frac{3}{2} \), where the parabola will touch the x-axis.
• The squared term indicates the parabola only touches the x-axis at this point, confirming a tangent at its vertex.
• Since the coefficient of \( (x - h)^2 \) is positive, the parabola opens upward.

This information helps in plotting a smooth curve that reflects the behavior of the quadratic function. Graphs can visually represent where the quadratic function changes direction and help identify other properties such as the maximum or minimum points.
With practice, graphing a parabola becomes an invaluable tool in understanding and interpreting quadratic equations.