Solve the following equation: 9x^2 - 39x - 30 = 0
Understand the Problem
The question is asking to solve the quadratic equation 9x^2 - 39x - 30 = 0. This involves finding the values of 'x' that satisfy the equation. Factoring, completing the square, or using the quadratic formula are common methods to solve this type of problem.
Answer
$x = -\frac{2}{3}, 5$
Answer for screen readers
$x = -\frac{2}{3}, 5$
Steps to Solve
- Simplify the equation
Divide the entire equation by 3 to simplify it:
$\frac{9x^2}{3} - \frac{39x}{3} - \frac{30}{3} = \frac{0}{3}$
$3x^2 - 13x - 10 = 0$
- Factor the quadratic equation
We want to find two numbers that multiply to $3 \times -10 = -30$ and add up to $-13$. These numbers are $-15$ and $2$. Rewrite the middle term using these numbers:
$3x^2 - 15x + 2x - 10 = 0$
- Factor by grouping
Factor out the greatest common factor from the first two terms and the last two terms:
$3x(x - 5) + 2(x - 5) = 0$
Now, factor out the common binomial factor $(x - 5)$:
$(3x + 2)(x - 5) = 0$
- Solve for x
Set each factor equal to zero and solve for $x$:
$3x + 2 = 0$ or $x - 5 = 0$
For $3x + 2 = 0$:
$3x = -2$ $x = -\frac{2}{3}$
For $x - 5 = 0$:
$x = 5$
$x = -\frac{2}{3}, 5$
More Information
The solutions to the quadratic equation $9x^2 - 39x - 30 = 0$ are $x = -\frac{2}{3}$ and $x=5$. These are the x-intercepts of the parabola represented by the quadratic equation.
Tips
A common mistake is not simplifying the equation in the beginning. Factoring $9x^2 - 39x - 30 = 0$ directly is possible, but more difficult than factoring $3x^2 - 13x - 10 = 0$. Another common mistake is making errors while factoring or applying the quadratic formula. Always double-check your work.
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