Question

Solve the quadratic by factoring.
3, x, squared, minus, 5, equals, minus, 5, x, minus, 3
3x
2
−5=−5x−3

Answer 1

To solve the quadratic equation \( 3x^2 - 5 = -5x - 3 \), let's first rearrange it into standard form, which is \( ax^2 + bx + c = 0 \).

1. Move all terms to one side of the equation:
\[
3x^2 + 5x - 5 + 3 = 0
\]

2. Simplify the equation:
\[
3x^2 + 5x - 2 = 0
\]

Next, we will factor the quadratic equation \( 3x^2 + 5x - 2 \).

To factor it, we look for two numbers that multiply to \( 3 \times -2 = -6 \) and add to \( 5 \). The two numbers that meet these criteria are \( 6 \) and \( -1 \).

We can rewrite the middle term (the \( 5x \)) using these numbers:
\[
3x^2 + 6x - x - 2 = 0
\]

Now we can group the terms:
\[
(3x^2 + 6x) + (-x - 2) = 0
\]
\[
3x(x + 2) - 1(x + 2) = 0
\]

Now factor out the common factor \( (x + 2) \):
\[
(3x - 1)(x + 2) = 0
\]

Now, set each factor to zero and solve for \( x \):
1. \( 3x - 1 = 0 \):
\[
3x = 1 \quad \Rightarrow \quad x = \frac{1}{3}
\]

2. \( x + 2 = 0 \):
\[
x = -2
\]

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