Solve the Quadratic Puzzle: 4/9x^2 + 8/3x + 4 = 0

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

When $ \Delta = 0 $, your quadratic has exactly one solution (called a double root). The parabola just touches the x-axis at one point instead of crossing it at two points.

Q: Why can't I just factor this equation instead?

With fractional coefficients like $ \frac{4}{9} $ and $ \frac{8}{3} $, factoring becomes very difficult. The quadratic formula works for any quadratic equation, especially with fractions!

Q: How do I calculate with these fractions without making errors?

Work step by step: $ \left(\frac{8}{3}\right)^2 = \frac{64}{9} $ and $ 4 \times \frac{4}{9} \times 4 = \frac{64}{9} $. Keep denominators consistent and double-check each calculation.

Q: Can I multiply the entire equation by 9 to clear fractions first?

Yes! Multiplying by 9 gives you $ 4x^2 + 24x + 36 = 0 $, which is easier to work with. Just remember to divide by 4 to get $ x^2 + 6x + 9 = 0 $.

Q: Why is the answer negative when we're squaring x?

Even though $ x^2 $ is always positive, the other terms in the equation can make the solution negative. The $ \frac{8}{3}x $ term dominates when x is negative!

Quadratic Formula with Zero Discriminant

Solve the following equation:

$\frac{4}{9}x^2+\frac{8}{3}x+4=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Multiply by 9 to eliminate the fraction

00:21 Simplify as much as possible

00:29 Identify the coefficients

00:44 Use the roots formula

00:57 Substitute appropriate values according to the given data and solve

01:23 Calculate the square and products

01:35 Square root of 0 is always equal to 0

01:41 When the root equals 0, there will be only one solution to the equation

01:58 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Solve the following equation:

$\frac{4}{9}x^2+\frac{8}{3}x+4=0$

Step-by-step solution

To solve the given quadratic equation $\frac{4}{9}x^2+\frac{8}{3}x+4=0$ , we will apply the Quadratic Formula. Let's outline the steps.

Step 1: Identify the coefficients.
$a = \frac{4}{9}, \, b = \frac{8}{3}, \, c = 4$ .
Step 2: Calculate the discriminant.
$\Delta = b^2 - 4ac = \left(\frac{8}{3}\right)^2 - 4\left(\frac{4}{9}\right)(4)$ .

Calculating $\Delta$ :
$\left(\frac{8}{3}\right)^2 = \frac{64}{9}$
$4 \times \frac{4}{9} \times 4 = \frac{64}{9}$ ,
so, $\Delta = \frac{64}{9} - \frac{64}{9} = 0$ .

Step 3: Use the Quadratic Formula. Since the discriminant is zero, there's a single solution.
The formula simplifies to $x = \frac{-b}{2a}$ .
Calculate $x = \frac{-\frac{8}{3}}{2 \times \frac{4}{9}} = \frac{-\frac{8}{3}}{\frac{8}{9}} = -3$ .

Therefore, the solution to the equation is $x = -3$ .

Final Answer

$x=-3$

Key Points to Remember

Essential concepts to master this topic

Rule: When discriminant equals zero, quadratic has one repeated solution
Technique: Calculate $\Delta = b^2 - 4ac = \frac{64}{9} - \frac{64}{9} = 0$
Check: Substitute $x = -3$ : $\frac{4}{9}(-3)^2 + \frac{8}{3}(-3) + 4 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Assuming zero discriminant means no solution
Don't think discriminant = 0 means no solution = wrong conclusion! Zero discriminant actually means one repeated root (double root). Always use the simplified formula $x = \frac{-b}{2a}$ when discriminant equals zero.

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 $\Delta = 0$ , your quadratic has exactly one solution (called a double root). The parabola just touches the x-axis at one point instead of crossing it at two points.

Why can't I just factor this equation instead?

With fractional coefficients like $\frac{4}{9}$ and $\frac{8}{3}$ , factoring becomes very difficult. The quadratic formula works for any quadratic equation, especially with fractions!

How do I calculate with these fractions without making errors?

Work step by step: $\left(\frac{8}{3}\right)^2 = \frac{64}{9}$ and $4 \times \frac{4}{9} \times 4 = \frac{64}{9}$ . Keep denominators consistent and double-check each calculation.

Can I multiply the entire equation by 9 to clear fractions first?

Yes! Multiplying by 9 gives you $4x^2 + 24x + 36 = 0$ , which is easier to work with. Just remember to divide by 4 to get $x^2 + 6x + 9 = 0$ .

Why is the answer negative when we're squaring x?

Even though $x^2$ is always positive, the other terms in the equation can make the solution negative. The $\frac{8}{3}x$ term dominates when x is negative!

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