Solve the Quadratic Equation: 9y² - 30y = -25

Q: What does it mean when the discriminant equals zero?

When $ b^2 - 4ac = 0 $, there's exactly one solution (a repeated root). The parabola touches the x-axis at only one point instead of crossing it twice.

Q: Do I always need to use the quadratic formula?

Not always! This equation could also be solved by factoring since $ 9y^2 - 30y + 25 = (3y - 5)^2 $. Try factoring first - if it doesn't work easily, use the formula.

Q: How do I know which method to choose?

Look for perfect square trinomials first (like this problem). If the discriminant is a perfect square, factoring might be easier. Otherwise, the quadratic formula always works!

Q: Why did we get a fraction as the answer?

Fractions are common solutions! Since we have $ y = \frac{30}{18} $, always simplify by dividing both numerator and denominator by their GCD (6) to get $ \frac{5}{3} $.

Q: How can I check if my answer is correct?

Substitute $ y = \frac{5}{3} $ into the original equation: $ 9(\frac{5}{3})^2 - 30(\frac{5}{3}) = 9(\frac{25}{9}) - 50 = 25 - 50 = -25 $ ✓

Quadratic Formula with Zero Discriminant

$9y^2-30y=-25$

❤️ Continue Your Math Journey!

We have hundreds of course questions with personalized recommendations + Account 100% premium

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:07 Let's find Y. Ready?

00:10 First, make sure one side of the equation equals zero. This sets the stage for solving it.

00:23 Now, let's factor 3, Y squared. We break it down step-by-step.

00:28 Next, think of 25 as 5 squared. This helps us later.

00:33 Now, factor 30 into two, three, and five. Great job so far!

00:42 We can use the quadratic formula to find our binomial expression. Keep going!

00:47 Next, we take the square root to remove the square. Almost there!

00:52 Now, isolate Y. You're doing great!

00:56 And there you have it! That's the solution to our problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

$9y^2-30y=-25$

Step-by-step solution

To solve the quadratic equation $9y^2 - 30y = -25$ , we first rewrite it in standard form:

$9y^2 - 30y + 25 = 0$ .

Identifying the coefficients, we have $a = 9$ , $b = -30$ , and $c = 25$ . We will apply the quadratic formula:

$y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .

First, compute the discriminant:

$b^2 - 4ac = (-30)^2 - 4 \cdot 9 \cdot 25 = 900 - 900 = 0$ .

Since the discriminant is zero, there is a single repeated root. Substituting back into the quadratic formula, we get:

$y = \frac{-(-30) \pm \sqrt{0}}{2 \cdot 9} = \frac{30}{18} = \frac{5}{3}$ .

Therefore, the solution to the equation is $y = \frac{5}{3}$ .

Final Answer

$y=\frac{5}{3}$

Key Points to Remember

Essential concepts to master this topic

Standard Form: Move all terms to one side first
Discriminant: Calculate $b^2 - 4ac = 900 - 900 = 0$
Check: Substitute $y = \frac{5}{3}$ back into original equation ✓

Common Mistakes

Avoid these frequent errors

Forgetting to rearrange equation to standard form first
Don't apply the quadratic formula to $9y^2 - 30y = -25$ directly = wrong coefficients! This gives incorrect values for a, b, and c, leading to completely wrong answers. Always rearrange to $ay^2 + by + c = 0$ form first.

Practice Quiz

Test your knowledge with interactive questions

a = Coefficient of x²

b = Coefficient of x

c = Coefficient of the independent number

what is the value of $$ a $$ in the equation

$$ y=3x-10+5x^2 $$

FAQ

Everything you need to know about this question

What does it mean when the discriminant equals zero?

When $b^2 - 4ac = 0$ , there's exactly one solution (a repeated root). The parabola touches the x-axis at only one point instead of crossing it twice.

Do I always need to use the quadratic formula?

Not always! This equation could also be solved by factoring since $9y^2 - 30y + 25 = (3y - 5)^2$ . Try factoring first - if it doesn't work easily, use the formula.

How do I know which method to choose?

Look for perfect square trinomials first (like this problem). If the discriminant is a perfect square, factoring might be easier. Otherwise, the quadratic formula always works!

Why did we get a fraction as the answer?

Fractions are common solutions! Since we have $y = \frac{30}{18}$ , always simplify by dividing both numerator and denominator by their GCD (6) to get $\frac{5}{3}$ .

How can I check if my answer is correct?

Substitute $y = \frac{5}{3}$ into the original equation: $9(\frac{5}{3})^2 - 30(\frac{5}{3}) = 9(\frac{25}{9}) - 50 = 25 - 50 = -25$ ✓

🌟 Unlock Your Math Potential

Get unlimited access to all 18 Solving Quadratic Equations questions, detailed video solutions, and personalized progress tracking.

📹

Unlimited Video Solutions

Step-by-step explanations for every problem

📊

Progress Analytics

Track your mastery across all topics

🚫

Ad-Free Learning

Focus on math without distractions

No credit card required • Cancel anytime