Question

Use the quadratic formula to determine the exact solutions to the equation.

6x2+4x−3=0



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x =
or x =

Answer 1

To solve the quadratic equation \(6x^2 + 4x - 3 = 0\) using the quadratic formula, we need to identify the coefficients \(a\), \(b\), and \(c\) from the standard form of a quadratic equation \(ax^2 + bx + c = 0\).

Here, we have:
- \(a = 6\)
- \(b = 4\)
- \(c = -3\)

The quadratic formula is given by:

\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]

Now we can substitute the values of \(a\), \(b\), and \(c\) into the formula.

1. Calculate the discriminant \(b^2 - 4ac\):

\[
b^2 = 4^2 = 16
\]
\[
4ac = 4 \cdot 6 \cdot (-3) = -72
\]
\[
b^2 - 4ac = 16 - (-72) = 16 + 72 = 88
\]

2. Now substitute \(b\) and the discriminant into the quadratic formula:

\[
x = \frac{-4 \pm \sqrt{88}}{2 \cdot 6}
\]

3. Simplify \(\sqrt{88}\):

\[
\sqrt{88} = \sqrt{4 \cdot 22} = 2\sqrt{22}
\]

4. Substitute this back into the formula:

\[
x = \frac{-4 \pm 2\sqrt{22}}{12}
\]

5. Simplify the fraction:

\[
x = \frac{-2 \pm \sqrt{22}}{6}
\]

Thus, the solutions are:

\[
x = \frac{-2 + \sqrt{22}}{6} \quad \text{or} \quad x = \frac{-2 - \sqrt{22}}{6}
\]

So, the exact solutions are:

\[
x = \frac{-2 + \sqrt{22}}{6} \quad \text{or} \quad x = \frac{-2 - \sqrt{22}}{6}
\]