91Ó°ÊÓ

When faced with a quadratic equation, one helpful tool is the quadratic formula. This formula gives us the roots (solutions) directly without requiring factorization. The quadratic formula is: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] The components of the formula come from the standard quadratic equation format, which is in the form of \[ ax^2 + bx + c = 0 \].

• a is the coefficient of x^2
• b is the coefficient of x
• c is the constant term

Plus-minus (\pm) indicates that we will have two solutions:

• One solution with addition
• One solution with subtraction

This is essential because quadratic equations represent parabolas and can intersect the x-axis at two points, one point, or not at all.

To use the quadratic formula, we first need to express our equation in the standard form \[ ax^2 + bx + c = 0 \]. In our exercise, we start with \[ 9x^2 + 6x = 1 \]. To convert this into the standard form, we subtract 1 from both sides to get: \[ 9x^2 + 6x - 1 = 0 \]. Now we can identify the coefficients:

• a = 9
• b = 6
• c = -1

Converting the equation into standard form is crucial because it allows us to accurately use the quadratic formula. Make sure your equation is in \[ ax^2 + bx + c = 0 \] form before proceeding.

Once we have the equation in standard form, solving it with the quadratic formula becomes a systematic task. Using the coefficients from our example (\[ a = 9 \],\[ b = 6 \],\[ c = -1 \]), we substitute them into the quadratic formula: \[ x = \frac{-6 \pm \sqrt{6^2 - 4\cdot 9 \cdot (-1)}}{2\cdot 9} \]. Next, we simplify the expression inside the square root (discriminant):

• \[ 6^2 = 36 \]
• \[ -4 \cdot 9 \cdot (-1) = 36 \]
• \[ 36 + 36 = 72 \]

So, our formula now looks like: \[ x = \frac{-6 \pm \sqrt{72}}{18} \]. Further simplifying, \[ \sqrt{72} = 6\sqrt{2} \], gives us: \[ x = \frac{-6 \pm 6\sqrt{2}}{18} \]. Then, factoring out 6 from the numerator and reducing the fraction, we get the final solutions: \[ x = \frac{-1 \pm \sqrt{2}}{3} \]. Breaking down the steps helps in understanding how each part of the quadratic formula functions and how to simplify the results correctly. Always ensure each step is carefully executed to avoid mistakes.