Question

solve 5x^2 +6x=3
quadratic formula

Answer 1

To solve the quadratic equation \(5x^2 + 6x = 3\) using the quadratic formula, we first need to rewrite the equation in standard form \(ax^2 + bx + c = 0\).

Starting with the original equation:

\[
5x^2 + 6x - 3 = 0
\]

Here, we identify the coefficients:
- \(a = 5\)
- \(b = 6\)
- \(c = -3\)

The quadratic formula is given by:

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

Now, we will calculate the discriminant \(b^2 - 4ac\):

\[
b^2 - 4ac = 6^2 - 4(5)(-3) = 36 + 60 = 96
\]

Next, we substitute the values of \(a\), \(b\), and the discriminant into the quadratic formula:

\[
x = \frac{{-6 \pm \sqrt{96}}}{2(5)}
\]

Now, simplify \(\sqrt{96}\):

\[
\sqrt{96} = \sqrt{16 \times 6} = 4\sqrt{6}
\]

So, we have:

\[
x = \frac{{-6 \pm 4\sqrt{6}}}{10}
\]

We can simplify this fraction:

\[
x = \frac{{-3 \pm 2\sqrt{6}}}{5}
\]

Therefore, the solutions to the equation \(5x^2 + 6x - 3 = 0\) are:

\[
x = \frac{{-3 + 2\sqrt{6}}}{5} \quad \text{and} \quad x = \frac{{-3 - 2\sqrt{6}}}{5}
\]