Solve for $g$: $9g^2 - 30g + 25 = 0$
Understand the Problem
The question asks us to solve for $g$ in the quadratic equation $9g^2 - 30g + 25 = 0$. This is a math problem that involves finding the value(s) of $g$ that satisfy the equation, and we can solve it by factoring the quadratic equation.
Answer
$g = \frac{5}{3}$
Answer for screen readers
$g = \frac{5}{3}$
Steps to Solve
- Factor the quadratic equation
We are given the quadratic equation $9g^2 - 30g + 25 = 0$.
We want to factor the quadratic expression on the left side. Notice that $9g^2 = (3g)^2$ and $25 = 5^2$. Also, $30g = 2(3g)(5)$. Therefore, the quadratic expression is a perfect square trinomial
$$ 9g^2 - 30g + 25 = (3g - 5)^2 $$
So the equation can be written as: $$ (3g-5)^2 = 0 $$
- Solve for g
Taking the square root of both sides:
$$ \sqrt{(3g-5)^2} = \sqrt{0} $$ $$ 3g - 5 = 0 $$
Add 5 to both sides:
$$ 3g = 5 $$
Divide by 3:
$$ g = \frac{5}{3} $$
$g = \frac{5}{3}$
More Information
The quadratic equation has only one solution, $g = \frac{5}{3}$. This is because the discriminant of the quadratic equation is zero. The discriminant is given by $b^2 - 4ac$, where $a = 9$, $b = -30$, and $c = 25$. In this case, the discriminant is $(-30)^2 - 4(9)(25) = 900 - 900 = 0$.
Tips
A common mistake is to incorrectly factor the quadratic equation. Another common mistake is to forget to take the square root of both sides after factoring. Also, some students may try to use the quadratic formula, which will also work but is not needed here since the equation can be easily factored.
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