Solve the Quadratic Equation with Negative Coefficient: -(x+3)²=4x

Q: Why do I get negative solutions when the original equation doesn't look negative?

The negative coefficient $ a = -1 $ in our quadratic creates a downward-opening parabola. This often results in negative x-intercepts, which are our solutions!

Q: Can I multiply the entire equation by -1 to make it easier?

Yes! Multiplying $ -x^2 - 10x - 9 = 0 $ by -1 gives $ x^2 + 10x + 9 = 0 $. This doesn't change the solutions but makes the quadratic formula easier to work with.

Q: How do I know if I distributed the negative sign correctly?

Check each term: $ -(x^2 + 6x + 9) $ becomes $ -x^2 - 6x - 9 $. Every term inside the parentheses gets multiplied by the negative sign!

Q: Why doesn't factoring work easily for this equation?

With $ a = -1 $, factoring becomes tricky. The quadratic formula is often the most reliable method for equations with negative leading coefficients.

Q: How can I check my discriminant calculation?

$ b^2 - 4ac = (-10)^2 - 4(-1)(-9) $$ = 100 - 36 = 64 $Since $ \sqrt{64} = 8 $, we get two real solutions!

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Find X

00:03 Use the shortened multiplication formulas

00:15 Substitute appropriate values according to the data and expand the brackets

00:51 Negative times positive is always negative

01:01 Arrange the equation so that one side equals 0

01:09 Collect like terms

01:24 Multiply by negative to convert from negative to positive

01:38 Examine the coefficients

01:50 Use the root formula

02:10 Substitute appropriate values and solve

02:27 Calculate the square and multiplications

02:43 Calculate the square root of 64

02:52 These are the 2 possible solutions (addition, subtraction)

03:06 And this is the solution to the problem

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Solve the following equation:

$-(x+3)^2=4x$

2

Step-by-step solution

To solve $-(x+3)^2 = 4x$ , follow these steps:

Step 1: Expand the left side: $-(x+3)^2 = -(x^2 + 6x + 9)$ .
Step 2: Distribute the negative sign: $-x^2 - 6x - 9$ .
Step 3: Set the equation by moving terms to the right: $-x^2 - 6x - 9 = 4x$ becomes $-x^2 - 6x - 9 - 4x = 0$ .
Step 4: Simplify to standard quadratic form: $-x^2 - 10x - 9 = 0$ .
Step 5: Applying the quadratic formula where $a = -1$ , $b = -10$ , $c = -9$ :
$x = \frac{-(-10) \pm \sqrt{(-10)^2 - 4 \cdot (-1) \cdot (-9)}}{2 \cdot (-1)}$ .
$x = \frac{10 \pm \sqrt{100 - 36}}{-2}$ .
$x = \frac{10 \pm \sqrt{64}}{-2}$ .
$x = \frac{10 \pm 8}{-2}$ .
Two solutions arise: $x = \frac{10 + 8}{-2} = -9$ and $x = \frac{10 - 8}{-2} = -1$ .

Therefore, the solutions are $x_1 = -1$ and $x_2 = -9$ .

Thus, the correct answer is $\mathbf{x_1=-1,x_2=-9}$ .

3

Final Answer

$x_1=-1,x_2=-9$

Key Points to Remember

Essential concepts to master this topic

Expansion: Distribute negative sign carefully when expanding $-(x+3)^2$
Standard Form: Move all terms to get $-x^2 - 10x - 9 = 0$
Verification: Substitute solutions back: $-(-1+3)^2 = -16$ and $4(-1) = -4$ ✓

Common Mistakes

Avoid these frequent errors

Forgetting to distribute the negative sign
Don't expand $-(x+3)^2$ as $-x^2 + 6x + 9$ = wrong signs! The negative affects all terms inside the parentheses. Always distribute the negative to get $-x^2 - 6x - 9$ .

Practice Quiz

Test your knowledge with interactive questions

a = coefficient of x²

b = coefficient of x

c = coefficient of the constant term

What is the value of $$ c $$ in the function $$ y=-x^2+25x $$ ?

FAQ

Everything you need to know about this question

Why do I get negative solutions when the original equation doesn't look negative?

+

The negative coefficient $a = -1$ in our quadratic creates a downward-opening parabola. This often results in negative x-intercepts, which are our solutions!

Can I multiply the entire equation by -1 to make it easier?

+

Yes! Multiplying $-x^2 - 10x - 9 = 0$ by -1 gives $x^2 + 10x + 9 = 0$ . This doesn't change the solutions but makes the quadratic formula easier to work with.

How do I know if I distributed the negative sign correctly?

+

Check each term: $-(x^2 + 6x + 9)$ becomes $-x^2 - 6x - 9$ . Every term inside the parentheses gets multiplied by the negative sign!

Why doesn't factoring work easily for this equation?

+

With $a = -1$ , factoring becomes tricky. The quadratic formula is often the most reliable method for equations with negative leading coefficients.

How can I check my discriminant calculation?

+

$b^2 - 4ac = (-10)^2 - 4(-1)(-9)$
$= 100 - 36 = 64$
Since $\sqrt{64} = 8$ , we get two real solutions!

🌟 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

Solve the Quadratic Equation with Negative Coefficient: -(x+3)²=4x

Quadratic Equations with Negative Leading Coefficients

❤️ Continue Your Math Journey!