Questions: Use the quadratic formula to solve the equation. 3x^2 + 4x - 3 = 0 The solution set is

Use the quadratic formula to solve the equation. 3x^2 + 4x - 3 = 0 The solution set is
Transcript text: Use the quadratic formula to solve the equation. \[ 3 x^{2}+4 x-3=0 \] The solution set is $\square$
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Solution

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Solution Steps

To solve the quadratic equation \(3x^2 + 4x - 3 = 0\) using the quadratic formula, we need to identify the coefficients \(a\), \(b\), and \(c\) from the equation \(ax^2 + bx + c = 0\). Then, we apply the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) to find the solutions.

Solution Approach
  1. Identify the coefficients \(a\), \(b\), and \(c\) from the equation.
  2. Compute the discriminant \(\Delta = b^2 - 4ac\).
  3. Use the quadratic formula to find the solutions for \(x\).
Step 1: Identify Coefficients

The given quadratic equation is \(3x^2 + 4x - 3 = 0\). Here, the coefficients are:

  • \(a = 3\)
  • \(b = 4\)
  • \(c = -3\)
Step 2: Calculate the Discriminant

We calculate the discriminant \(\Delta\) using the formula: \[ \Delta = b^2 - 4ac \] Substituting the values: \[ \Delta = 4^2 - 4 \cdot 3 \cdot (-3) = 16 + 36 = 52 \]

Step 3: Apply the Quadratic Formula

Using the quadratic formula \(x = \frac{-b \pm \sqrt{\Delta}}{2a}\), we find the solutions: \[ x_1 = \frac{-4 + \sqrt{52}}{2 \cdot 3} \quad \text{and} \quad x_2 = \frac{-4 - \sqrt{52}}{2 \cdot 3} \] Calculating these values: \[ x_1 = \frac{-4 + 2\sqrt{13}}{6} = \frac{-2 + \sqrt{13}}{3} \approx 0.5352 \] \[ x_2 = \frac{-4 - 2\sqrt{13}}{6} = \frac{-2 - \sqrt{13}}{3} \approx -1.8685 \]

Final Answer

The solutions to the equation \(3x^2 + 4x - 3 = 0\) are: \[ \boxed{x_1 = \frac{-2 + \sqrt{13}}{3}, \quad x_2 = \frac{-2 - \sqrt{13}}{3}} \]

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