Problem 16 - Beginning and Intermediate Algebra

Question

Use the quadratic formula to solve each equation. (All solutions for these equations are real numbers.) $$ 9 x^{2}+6 x=1 $$

Step-by-Step Solution

Answer

The solutions are \( x = \frac{1 - \sqrt{2}}{3} \) and \( x = \frac{1 + \sqrt{2}}{3} \).

Step 1: Write the equation in standard form

The given equation is: 9x^{2} + 6x = 1. To use the quadratic formula, the equation needs to be in the standard form, which is ax^{2} + bx + c = 0. Subtract 1 from both sides to achieve this: 9x^{2} + 6x - 1 = 0.

Step 2: Identify coefficients a, b, and c

From the equation 9x^{2} + 6x - 1 = 0, identify the coefficients: a = 9, b = 6, c = -1.

Step 3: Recall the quadratic formula

The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \]

Step 4: Substitute the coefficients into the formula

Substitute a = 9, b = 6, and c = -1 into the quadratic formula: \[ x = \frac{-(6) \pm \sqrt{(6)^{2} - 4(9)(-1)}}{2(9)} \] This simplifies to: \[ x = \frac{-6 \pm \sqrt{36 + 36}}{18} \] \[ x = \frac{-6 \pm \sqrt{72}}{18} \]

Step 5: Simplify the square root

Simplify \( \sqrt{72} \), noting that \( \sqrt{72} = \sqrt{36 \cdot 2} = 6 \sqrt{2} \), thus: \[ x = \frac{-6 \pm 6 \sqrt{2}}{18} \]

Step 6: Simplify the fraction

Factor out the common term in the numerator and denominator: \[ x = \frac{-6(1 \pm \sqrt{2})}{18} \] \[ x = \frac{1 \pm \sqrt{2}}{3} \]

Step 7: Write the solutions

Thus, the solutions to the quadratic equation are: \[ x = \frac{1 - \sqrt{2}}{3} \] and \[ x = \frac{1 + \sqrt{2}}{3} \]

Key Concepts

standard form of quadratic equation

To solve a quadratic equation using the quadratic formula, the equation must be in its standard form. The standard form of a quadratic equation is given by: \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are coefficients, where:

• \(a\) represents the coefficient of \(x^2\)
• \(b\) represents the coefficient of \(x\)
• \(c\) is the constant term

For example, the equation \(9x^{2} + 6x = 1\) must be rearranged into the form \(9x^{2} + 6x - 1 = 0\) by subtracting 1 from both sides. This makes it easy to identify the coefficients: \(a = 9\), \(b = 6\), and \(c = -1\). This form is crucial because it sets up the equation for the use of the quadratic formula.

solving quadratics

Quadratic equations can be solved by different methods such as factoring, completing the square, or using the quadratic formula. The quadratic formula is a powerful tool because it works for any quadratic equation. It is given by: \[ x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a} \] To solve the given equation \( 9x^{2} + 6x - 1 = 0\), follow these steps:

• Identify coefficients: \( a = 9 \), \( b = 6 \), \( c = -1 \)
• Substitute these into the quadratic formula: \[ x = \frac{-(6) \pm \sqrt{(6)^{2} - 4(9)(-1)}}{2(9)} \]
• Simplify the expression under the square root: \[ x = \frac{-6 \pm \sqrt{36 + 36}}{18} \]
• This yields: \[ x = \frac{-6 \pm \sqrt{72}}{18} \]

The next step is to simplify the square root.

simplifying square roots

Simplifying square roots is essential for making solutions more understandable. To simplify \(\sqrt{72}\):

• Identify factors: \(72 = 36 \cdot 2\)
• Since \(36\) is a perfect square, \(\sqrt{36 \cdot 2} = 6\sqrt{2}\)

Thus, our solution becomes: \[ x = \frac{-6 \pm 6\sqrt{2}}{18} \] To further simplify:

• Factor out the common term in the numerator and denominator: \[ x = \frac{-6(1 \pm \sqrt{2})}{18} \]
• Divide both numerator and denominator by 6: \[ x = \frac{1 \pm \sqrt{2}}{3} \]

The final solutions are: \[ x = \frac{1 - \sqrt{2}}{3} \] and \[ x = \frac{1 + \sqrt{2}}{3} \] Breaking down each part of the process helps clarify the method and ensures accurate results.