Problem 43 - Algebra and Trigonometry Real Mathematics, Real People

Question

Solve the quadratic equation by completing the square. Verify your answer graphically. $$9 x^{2}-18 x+3=0$$

Step-by-Step Solution

Answer

The solutions of the given quadratic equation are \(x = 1 + \sqrt{2/3}\) and \(x = 1 - \sqrt{2/3}\)

Step 1: Express the Quadratic Equation

We write down the given equation and express it in the standard form of a quadratic equation. The equation is \(9x^2 - 18x + 3 = 0\).

Step 2: Solving for x by Completing the Square

In this step, the aim is to make the left-hand side of the equation a perfect square trinomial. First we divide the equation by the coefficient of \(x^2\), which is 9: \(x^2 - 2x + 1/3 = 0\). Then, we rewrite the quadratic and constant terms and move the constant term to the other side of the equation to give \(x^2 - 2x = -1/3\). Adding 1 (which is \((-b/2)^2\)) to both sides to complete the square, we obtain \((x - 1)^2 = -1/3 + 1\), Simplifying gives \((x - 1)^2 = 2/3\). Our equation has now been transformed into a square on the left side.

Step 3: Solve for x

We then take the square root of both sides of the equation to find the values of \(x\): \(x - 1 = \pm \sqrt{2/3}\), which gives two solutions, \(x = 1 + \sqrt{2/3}\) and \(x = 1 - \sqrt{2/3}\).

Step 4: Verification of Solutions Graphically

To verify the solutions graphically, we need to sketch the graph of the equation. We won't be representing it here, but the graph of the equation \(y = 9x^2 - 18x + 3\) should touch the x-axis at the solutions \(x = 1 + \sqrt{2/3}\) and \(x = 1 - \sqrt{2/3}\), thus verifying them.

Key Concepts

Completing the Square

Completing the square is a handy method for solving quadratic equations by transforming them into a perfect square trinomial. This approach makes equations easier to solve because perfect squares are simpler to manipulate. First, ensure your quadratic equation is in the standard form:

• For instance, with the equation \(9x^2 - 18x + 3 = 0\), divide every term by 9 to make the coefficient of \(x^2\) equal to 1.
• This gives \(x^2 - 2x + \frac{1}{3} = 0\).
• Next, isolate the \(x^2\) and \(x\) terms on one side: \(x^2 - 2x = -\frac{1}{3}\).
• We then complete the square by adding and subtracting the same number: here, add \(1\), resulting in the form \((x-1)^2\).

This leads to \((x - 1)^2 = \frac{2}{3}\), where the left side is now a perfect square. The equation is simpler because it directly gives a solvable form.

Solving Equations

Once you have a perfect square, solving the equation becomes straightforward. Solve the simplified equation \((x - 1)^2 = \frac{2}{3}\) by taking the square root of both sides:

• By doing this, you get: \(x - 1 = \pm \sqrt{\frac{2}{3}}\).
• This implies two potential solutions for \(x\): \(x = 1 + \sqrt{\frac{2}{3}}\) and \(x = 1 - \sqrt{\frac{2}{3}}\).

Taking square roots can yield both positive and negative roots, which align with the dual nature of quadratic solutions. Remember, each solution represents a point at which the equation touches or intersects the x-axis. This methodology not only simplifies solving quadratic equations but also leads effortlessly into graphical interpretations.

Graphical Verification

Graphical verification is a powerful tool to confirm solutions visually. Once you have solutions from completing the square, use them to verify graphically by plotting the quadratic equation. When you plot an equation like \(y = 9x^2 - 18x + 3\):

• Identify the solutions calculated, \(x = 1 + \sqrt{\frac{2}{3}}\) and \(x = 1 - \sqrt{\frac{2}{3}}\), which are the x-intercepts of the graph.
• The graph should intersect the x-axis at these points.
• A smooth, U-shaped curve indicates a parabola opening upwards, as expected from a positive \(x^2\) coefficient.

When the curve touches the x-axis at the solutions, this verifies that the solutions you computed accurately represent the places where the equation is zero.

Perfect Square Trinomial

Understanding perfect square trinomials is crucial when completing the square. A perfect square trinomial is an expression of the form \(a^2 - 2ab + b^2\). This can be rewritten as \((a - b)^2\).

• The expression \(x^2 - 2x + \frac{1}{3}\) transforms into \((x-1)^2\) after completing the square.
• The goal is to reshape part of the quadratic into this structured form because it's easily decomposable back into linear factors \((x-b)\).
• Knowing how to identify or create these trinomial forms equips you to solve quadratic equations quickly and efficiently.

Recognizing and constructing perfect square trinomials streamlines the solving process, facilitates easier calculations, and supports error checking within computations.