Solve the Quadratic Equation: -4x² + 96x - 576 = 0

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

When $ b^2 - 4ac = 0 $, the parabola touches the x-axis at exactly one point. This means there's one repeated root - the same solution occurring twice mathematically.

Q: Why don't I need the ± symbol when discriminant is zero?

The ± symbol accounts for $ \pm\sqrt{b^2-4ac} $. When the discriminant is zero, $ \sqrt{0} = 0 $, so both the + and - give the same answer!

Q: Can I factor this equation instead of using the quadratic formula?

Yes! You can factor out -4 first: $ -4(x^2 - 24x + 144) = 0 $. Then $ x^2 - 24x + 144 = (x-12)^2 $, giving $ x = 12 $.

Q: How do I know if I calculated the discriminant correctly?

Double-check each step: $ 96^2 = 9216 $ and $ 4(-4)(-576) = 9216 $. So $ 9216 - 9216 = 0 $. Always be careful with negative signs!

Q: What if I got a different answer like x = 4?

Check by substituting: $ -4(4)^2 + 96(4) - 576 = -64 + 384 - 576 = -256 \neq 0 $. This shows x = 4 is not correct. Only x = 12 works.

Quadratic Equations with Zero Discriminant

Solve the following equation:

$-4x^2+96x-576=0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Pay attention to the coefficients

00:10 Use the roots formula

00:29 Substitute appropriate values according to the given data and solve for X

00:51 Calculate the products and the square

01:17 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:

$-4x^2+96x-576=0$

Step-by-step solution

To solve this quadratic equation $-4x^2 + 96x - 576 = 0$ , we will apply the quadratic formula:

The general form of a quadratic equation is $ax^2 + bx + c = 0$ .
For this equation, the coefficients are $a = -4$ , $b = 96$ , and $c = -576$ .
The quadratic formula states: $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ .
First, compute the discriminant: $b^2 - 4ac = 96^2 - 4(-4)(-576)$ .
Calculate $96^2 = 9216$ and $4(-4)(-576) = 9216$ as well.
Thus, the discriminant is $9216 - 9216 = 0$ .
A discriminant of zero indicates that there is exactly one real solution (a repeated root).
Now substitute into the quadratic formula: $x = \frac{-96 \pm \sqrt{0}}{2(-4)}$ .
This simplifies to $x = \frac{-96}{-8}$ .
Further simplification gives $x = 12$ .

Therefore, the solution to the equation is $x = 12$ , which corresponds to answer choice 2.

Final Answer

$x=12$

Key Points to Remember

Essential concepts to master this topic

Discriminant Rule: When $b^2 - 4ac = 0$ , there is exactly one solution
Formula Method: Use $x = \frac{-b}{2a}$ when discriminant equals zero
Verification: Substitute $x = 12$ back: $-4(12)^2 + 96(12) - 576 = 0$ ✓

Common Mistakes

Avoid these frequent errors

Assuming there are two different solutions when discriminant is zero
Don't write x₁ = 12 and x₂ = 12 as two separate answers = confusion about repeated roots! When the discriminant equals zero, there's only ONE unique solution that occurs twice. Always recognize that zero discriminant means exactly one solution (a repeated root).

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 $b^2 - 4ac = 0$ , the parabola touches the x-axis at exactly one point. This means there's one repeated root - the same solution occurring twice mathematically.

Why don't I need the ± symbol when discriminant is zero?

The ± symbol accounts for $\pm\sqrt{b^2-4ac}$ . When the discriminant is zero, $\sqrt{0} = 0$ , so both the + and - give the same answer!

Can I factor this equation instead of using the quadratic formula?

Yes! You can factor out -4 first: $-4(x^2 - 24x + 144) = 0$ . Then $x^2 - 24x + 144 = (x-12)^2$ , giving $x = 12$ .

How do I know if I calculated the discriminant correctly?

Double-check each step: $96^2 = 9216$ and $4(-4)(-576) = 9216$ . So $9216 - 9216 = 0$ . Always be careful with negative signs!

What if I got a different answer like x = 4?

Check by substituting: $-4(4)^2 + 96(4) - 576 = -64 + 384 - 576 = -256 \neq 0$ . This shows x = 4 is not correct. Only x = 12 works.

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